💡Okay, so I have this problem where I need to find the probability that the slope of the line connecting a randomly chosen point Q inside the unit square to the point (3/4, 1/4) is greater than or equal to 1. The unit square has vertices at (0,0), (1,0), (1,1), and (0,1). I need to express this probability as a fraction m/n where m and n are coprime, and then find m + n.First, I should recall that the slope between two points (x1, y1) and (x2, y2) is given by (y2 - y1)/(x2 - x1). In this case, one point is fixed at (3/4, 1/4), and the other point Q is (x, y) where x and y are between 0 and 1. So, the slope of the line connecting Q and (3/4, 1/4) is (y - 1/4)/(x - 3/4).We want this slope to be greater than or equal to 1. So, the inequality is:(y - 1/4)/(x - 3/4) ≥ 1I need to solve this inequality for y in terms of x to figure out the region within the unit square where this condition holds.Multiplying both sides by (x - 3/4) gives:y - 1/4 ≥ x - 3/4But wait, I have to be careful here because if (x - 3/4) is negative, the inequality sign will flip when I multiply both sides. So, I should consider two cases: when (x - 3/4) is positive and when it's negative.Case 1: x - 3/4 > 0, which implies x > 3/4.In this case, multiplying both sides by (x - 3/4) doesn't change the inequality sign:y - 1/4 ≥ x - 3/4Adding 1/4 to both sides:y ≥ x - 3/4 + 1/4Simplify:y ≥ x - 1/2So, in this case, the condition is y ≥ x - 1/2.Case 2: x - 3/4 < 0, which implies x < 3/4.Here, multiplying both sides by (x - 3/4) will flip the inequality sign:y - 1/4 ≤ x - 3/4Adding 1/4 to both sides:y ≤ x - 3/4 + 1/4Simplify:y ≤ x - 1/2So, in this case, the condition is y ≤ x - 1/2.But wait, x is less than 3/4, so x - 1/2 would be less than 3/4 - 1/2 = 1/4. Since y is between 0 and 1, let's see if this inequality makes sense.If x < 3/4, then x - 1/2 could be negative, which would mean y ≤ some negative number. But y is always ≥ 0, so in that case, there are no solutions because y can't be less than or equal to a negative number. So, actually, for x < 3/4, the inequality y ≤ x - 1/2 would only have solutions when x - 1/2 ≥ 0, which is when x ≥ 1/2.Wait, so for x between 1/2 and 3/4, x - 1/2 is between 0 and 1/4. So, in this interval, y ≤ x - 1/2 would be a valid condition because y can be between 0 and x - 1/2.But hold on, let me double-check. If x is between 1/2 and 3/4, then x - 1/2 is between 0 and 1/4. So, y has to be less than or equal to that. So, in this region, the area where y ≤ x - 1/2 is a region below the line y = x - 1/2.But when x is less than 1/2, x - 1/2 is negative, so y ≤ x - 1/2 would imply y ≤ negative number, which is impossible because y is at least 0. So, for x < 1/2, there are no points that satisfy the inequality.So, summarizing:- For x < 1/2: No solutions.- For 1/2 ≤ x < 3/4: y ≤ x - 1/2.- For x ≥ 3/4: y ≥ x - 1/2.So, the region where the slope is ≥ 1 consists of two parts:1. A triangular region from x = 1/2 to x = 3/4 where y is between 0 and x - 1/2.2. Another region from x = 3/4 to x = 1 where y is between x - 1/2 and 1.Wait, actually, let me visualize this. The line y = x - 1/2 intersects the unit square at two points. Let me find those intersection points.When x = 1/2, y = 0. So, one point is (1/2, 0).When y = 0, x = 1/2, which we already have.When x = 1, y = 1 - 1/2 = 1/2. So, the other point is (1, 1/2).So, the line y = x - 1/2 goes from (1/2, 0) to (1, 1/2). So, in the unit square, this line starts at the midpoint of the bottom side and goes up to the midpoint of the right side.So, above this line, for x ≥ 3/4, y ≥ x - 1/2 is above the line, which is a triangle from (3/4, 1/4) to (1, 1/2) to (1, 1). Wait, no, actually, the region y ≥ x - 1/2 for x ≥ 3/4 is above the line y = x - 1/2, which is a triangle with vertices at (3/4, 1/4), (1, 1/2), and (1, 1).Similarly, for 1/2 ≤ x ≤ 3/4, the region y ≤ x - 1/2 is below the line y = x - 1/2, which is a triangle with vertices at (1/2, 0), (3/4, 1/4), and (1, 1/2).Wait, but actually, when x is between 1/2 and 3/4, y has to be less than or equal to x - 1/2, which is a line from (1/2, 0) to (3/4, 1/4). So, the region below this line is a triangle with vertices at (1/2, 0), (3/4, 1/4), and (1/2, 0) again? Wait, no, that doesn't make sense.Wait, let me think again. For x between 1/2 and 3/4, y ≤ x - 1/2. So, at x = 1/2, y ≤ 0, which is just the point (1/2, 0). At x = 3/4, y ≤ 1/4. So, the region is a trapezoid? Or is it a triangle?Wait, actually, it's a triangle because it's bounded by y = 0, x = 1/2, and y = x - 1/2.Wait, no, when x is between 1/2 and 3/4, y is between 0 and x - 1/2. So, it's a region that starts at (1/2, 0) and goes up to (3/4, 1/4). So, it's a triangular region with vertices at (1/2, 0), (3/4, 1/4), and (1/2, 0). Wait, that can't be right because that would just be a line.Wait, maybe I'm overcomplicating. Let me sketch it mentally.The line y = x - 1/2 goes from (1/2, 0) to (1, 1/2). So, above this line is the region y ≥ x - 1/2, and below is y ≤ x - 1/2.But for x < 1/2, y ≤ x - 1/2 is impossible because y can't be negative. So, the only regions where the inequality holds are:1. For x ≥ 3/4, y ≥ x - 1/2.2. For 1/2 ≤ x ≤ 3/4, y ≤ x - 1/2.So, these are two separate regions.Let me calculate the area of each region.First, for x ≥ 3/4, y ≥ x - 1/2.This is a triangular region with vertices at (3/4, 1/4), (1, 1/2), and (1, 1).Wait, let's see:At x = 3/4, y = 3/4 - 1/2 = 1/4.At x = 1, y = 1 - 1/2 = 1/2.So, the line y = x - 1/2 from (3/4, 1/4) to (1, 1/2). But the region y ≥ x - 1/2 for x ≥ 3/4 is above this line, so it's a triangle with vertices at (3/4, 1/4), (1, 1/2), and (1, 1).Wait, no, because at x = 1, y can go up to 1, but the line y = x - 1/2 at x = 1 is y = 1/2. So, the region y ≥ 1/2 for x = 1 is a vertical line segment from (1, 1/2) to (1, 1). But actually, it's a region bounded by x from 3/4 to 1, y from x - 1/2 to 1.So, this is a trapezoid, but actually, since the top boundary is y = 1 and the bottom boundary is y = x - 1/2, which is a line.Wait, maybe it's easier to compute the area by integrating.The area for x from 3/4 to 1 is the integral from x = 3/4 to x = 1 of (1 - (x - 1/2)) dx.Simplify the integrand: 1 - x + 1/2 = (3/2 - x).So, the integral becomes ∫ from 3/4 to 1 of (3/2 - x) dx.Compute the integral:The antiderivative of 3/2 is (3/2)x, and the antiderivative of -x is -(1/2)x².So, evaluating from 3/4 to 1:At x = 1: (3/2)(1) - (1/2)(1)² = 3/2 - 1/2 = 1.At x = 3/4: (3/2)(3/4) - (1/2)(9/16) = (9/8) - (9/32) = (36/32 - 9/32) = 27/32.So, the area is 1 - 27/32 = 5/32.Wait, that seems small. Let me check my calculations.Wait, the integral from 3/4 to 1 of (3/2 - x) dx.Compute:∫(3/2 - x) dx = (3/2)x - (1/2)x² evaluated from 3/4 to 1.At x = 1: (3/2)(1) - (1/2)(1) = 3/2 - 1/2 = 1.At x = 3/4: (3/2)(3/4) - (1/2)(9/16) = 9/8 - 9/32.Convert to common denominator:9/8 = 36/32, so 36/32 - 9/32 = 27/32.So, the area is 1 - 27/32 = 5/32.Okay, so that's 5/32.Now, the other region is for 1/2 ≤ x ≤ 3/4, y ≤ x - 1/2.This is the region below the line y = x - 1/2 from x = 1/2 to x = 3/4.So, the area is the integral from x = 1/2 to x = 3/4 of (x - 1/2) dx.Compute this integral:∫(x - 1/2) dx from 1/2 to 3/4.Antiderivative is (1/2)x² - (1/2)x.Evaluate at x = 3/4:(1/2)(9/16) - (1/2)(3/4) = 9/32 - 3/8 = 9/32 - 12/32 = -3/32.Evaluate at x = 1/2:(1/2)(1/4) - (1/2)(1/2) = 1/8 - 1/4 = -1/8.So, the area is (-3/32) - (-1/8) = (-3/32) + 4/32 = 1/32.Wait, that can't be right because the area can't be negative. Wait, no, the integral gives the net area, but since the function is positive in this interval, the area should be positive.Wait, let me recast the integral:∫ from 1/2 to 3/4 of (x - 1/2) dx.Let me make a substitution: let u = x - 1/2, then du = dx, and when x = 1/2, u = 0; when x = 3/4, u = 1/4.So, the integral becomes ∫ from 0 to 1/4 of u du = (1/2)u² evaluated from 0 to 1/4 = (1/2)(1/16) - 0 = 1/32.Yes, that's correct. So, the area is 1/32.So, the total area where the slope is ≥ 1 is the sum of these two regions: 5/32 + 1/32 = 6/32 = 3/16.Wait, but earlier I thought it was 1/8, but now I'm getting 3/16. Hmm, I must have made a mistake somewhere.Wait, let me double-check the regions.When x ≥ 3/4, the region is above y = x - 1/2, which is a triangle with vertices at (3/4, 1/4), (1, 1/2), and (1, 1). The area of this triangle can be calculated as follows:The base is from x = 3/4 to x = 1, which is 1/4. The height is from y = 1/2 to y = 1, which is 1/2. But wait, actually, the triangle is a right triangle with legs of length 1/4 and 1/2.Wait, no, the triangle is not a right triangle. Let me think.The line y = x - 1/2 goes from (3/4, 1/4) to (1, 1/2). The region above this line within the unit square is a trapezoid, but actually, it's a triangle because it's bounded by y = 1 and x = 1.Wait, perhaps it's better to use the formula for the area of a triangle given three points.The three vertices are (3/4, 1/4), (1, 1/2), and (1, 1).Using the formula for the area of a triangle with vertices (x1, y1), (x2, y2), (x3, y3):Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|Plugging in the points:x1 = 3/4, y1 = 1/4x2 = 1, y2 = 1/2x3 = 1, y3 = 1Area = |( (3/4)(1/2 - 1) + 1(1 - 1/4) + 1(1/4 - 1/2) ) / 2 |Simplify each term:(3/4)(-1/2) = -3/81*(3/4) = 3/41*(-1/4) = -1/4So, summing these: -3/8 + 3/4 - 1/4Convert to eighths:-3/8 + 6/8 - 2/8 = ( -3 + 6 - 2 ) / 8 = 1/8Take absolute value and divide by 2:|1/8| / 2 = 1/16Wait, that's different from the integral result. Hmm.Wait, maybe I made a mistake in the formula.Wait, the formula is:Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) / 2|So, plugging in:(3/4)(1/2 - 1) + 1(1 - 1/4) + 1(1/4 - 1/2)= (3/4)(-1/2) + 1*(3/4) + 1*(-1/4)= -3/8 + 3/4 - 1/4Convert to eighths:-3/8 + 6/8 - 2/8 = ( -3 + 6 - 2 ) / 8 = 1/8So, the area is |1/8| / 2 = 1/16.Wait, but earlier, when I integrated, I got 5/32 for this region. So, which one is correct?Wait, 1/16 is 2/32, and 5/32 is larger. Hmm.Wait, maybe I made a mistake in the integral.Let me recast the integral for x from 3/4 to 1, y from x - 1/2 to 1.So, the area is ∫ from 3/4 to 1 of (1 - (x - 1/2)) dx = ∫ from 3/4 to 1 of (3/2 - x) dx.Compute the integral:Antiderivative is (3/2)x - (1/2)x².Evaluate at x = 1: (3/2)(1) - (1/2)(1) = 3/2 - 1/2 = 1.Evaluate at x = 3/4: (3/2)(3/4) - (1/2)(9/16) = 9/8 - 9/32 = (36/32 - 9/32) = 27/32.So, the area is 1 - 27/32 = 5/32.But according to the triangle area formula, it's 1/16.Wait, 5/32 is approximately 0.15625, and 1/16 is 0.0625. These are different.Wait, maybe the triangle area formula was misapplied because the three points don't form a triangle in the way I thought.Wait, the three points are (3/4, 1/4), (1, 1/2), and (1, 1). Let me plot these points.(3/4, 1/4) is on the line y = x - 1/2.(1, 1/2) is also on that line.(1, 1) is the top-right corner.So, the region above y = x - 1/2 from x = 3/4 to x = 1 is bounded by the line from (3/4, 1/4) to (1, 1/2), and then up to (1, 1). So, it's a trapezoid with vertices at (3/4, 1/4), (1, 1/2), (1, 1), and (3/4, 1/4). Wait, no, that's not a trapezoid because (3/4, 1/4) is connected to (1, 1/2), which is connected to (1, 1), and then back to (3/4, 1/4). So, it's a triangle with vertices at (3/4, 1/4), (1, 1/2), and (1, 1).Wait, but when I calculated the area using the formula, I got 1/16, but integrating gave me 5/32. There's a discrepancy here.Wait, perhaps I made a mistake in the triangle area formula. Let me recalculate.Using the formula:Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) / 2|Plugging in:x1 = 3/4, y1 = 1/4x2 = 1, y2 = 1/2x3 = 1, y3 = 1So,Area = |( (3/4)(1/2 - 1) + 1(1 - 1/4) + 1(1/4 - 1/2) ) / 2 |= |( (3/4)(-1/2) + 1*(3/4) + 1*(-1/4) ) / 2 |= |( -3/8 + 3/4 - 1/4 ) / 2 |Convert to eighths:-3/8 + 6/8 - 2/8 = ( -3 + 6 - 2 ) / 8 = 1/8So, Area = |1/8| / 2 = 1/16.Wait, that's 1/16, but the integral gave me 5/32. These can't both be correct.Wait, perhaps the triangle area formula is incorrect because the points are not in the correct order or something.Alternatively, maybe I should use the shoelace formula correctly.Let me list the points in order:(3/4, 1/4), (1, 1/2), (1, 1).Using shoelace formula:Area = 1/2 | (x1y2 + x2y3 + x3y1) - (y1x2 + y2x3 + y3x1) |Compute:x1y2 = (3/4)(1/2) = 3/8x2y3 = (1)(1) = 1x3y1 = (1)(1/4) = 1/4Sum: 3/8 + 1 + 1/4 = 3/8 + 8/8 + 2/8 = 13/8Now, y1x2 = (1/4)(1) = 1/4y2x3 = (1/2)(1) = 1/2y3x1 = (1)(3/4) = 3/4Sum: 1/4 + 1/2 + 3/4 = (1 + 2 + 3)/4 = 6/4 = 3/2So, Area = 1/2 |13/8 - 3/2| = 1/2 |13/8 - 12/8| = 1/2 |1/8| = 1/16.So, the area is indeed 1/16.But wait, earlier, when I integrated, I got 5/32. So, which one is correct?Wait, 1/16 is 2/32, and 5/32 is larger. So, there must be a mistake in the integration.Wait, let me check the integral again.The region is for x from 3/4 to 1, y from x - 1/2 to 1.So, the area is ∫ from 3/4 to 1 of (1 - (x - 1/2)) dx = ∫ from 3/4 to 1 of (3/2 - x) dx.Compute:∫(3/2 - x) dx = (3/2)x - (1/2)x²Evaluate at x = 1: (3/2)(1) - (1/2)(1) = 3/2 - 1/2 = 1.Evaluate at x = 3/4: (3/2)(3/4) - (1/2)(9/16) = 9/8 - 9/32 = (36/32 - 9/32) = 27/32.So, the area is 1 - 27/32 = 5/32.Wait, but according to the shoelace formula, it's 1/16. So, which one is correct?Wait, perhaps I'm misunderstanding the region. The integral gives the area under the curve y = 3/2 - x from x = 3/4 to x = 1, but the actual region is bounded by y = x - 1/2 and y = 1.Wait, no, the integral is correct because it's integrating the vertical distance from y = x - 1/2 to y = 1, which is (1 - (x - 1/2)) = 3/2 - x.So, the integral should be correct, giving 5/32.But the shoelace formula gave me 1/16 for the triangle with vertices at (3/4, 1/4), (1, 1/2), (1, 1). So, which one is correct?Wait, perhaps the shoelace formula is correct because it's a triangle, and the integral is over a different region.Wait, no, the integral is over the same region as the triangle. So, why the discrepancy?Wait, maybe I made a mistake in the shoelace formula.Wait, let me recast the points:(3/4, 1/4), (1, 1/2), (1, 1).Plotting these, the triangle is formed by these three points.Using shoelace formula:List the points in order:(3/4, 1/4), (1, 1/2), (1, 1), (3/4, 1/4).Compute the sum of x_i y_{i+1}:(3/4)(1/2) + (1)(1) + (1)(1/4) = 3/8 + 1 + 1/4 = 3/8 + 8/8 + 2/8 = 13/8.Compute the sum of y_i x_{i+1}:(1/4)(1) + (1/2)(1) + (1)(3/4) = 1/4 + 1/2 + 3/4 = (1 + 2 + 3)/4 = 6/4 = 3/2.So, area = 1/2 |13/8 - 3/2| = 1/2 |13/8 - 12/8| = 1/2 * 1/8 = 1/16.So, shoelace formula says 1/16.But the integral says 5/32.Wait, 1/16 is 2/32, and 5/32 is larger. So, which one is correct?Wait, perhaps the integral is incorrect because the region is not a straight vertical slice but a triangle.Wait, no, the integral is correct because it's integrating the vertical distance from y = x - 1/2 to y = 1, which is a straight line, so the area should be a trapezoid, but in this case, since the top boundary is flat at y = 1, it's a trapezoid with one side slanting.Wait, but according to the shoelace formula, it's a triangle with area 1/16.Wait, maybe I'm confusing the regions.Wait, let me think differently. The line y = x - 1/2 intersects the square at (1/2, 0) and (1, 1/2). So, the region where y ≥ x - 1/2 is above this line.So, for x from 1/2 to 1, y ≥ x - 1/2.But when x is from 1/2 to 3/4, y ≥ x - 1/2 is a region that starts at (1/2, 0) and goes up to (3/4, 1/4). But wait, no, because for x from 1/2 to 3/4, y has to be ≥ x - 1/2, but y can't be negative, so it's from y = x - 1/2 to y = 1.Wait, no, actually, for x from 1/2 to 3/4, y ≥ x - 1/2 is a region that starts at (1/2, 0) and goes up to (3/4, 1/4). But above that line, y can go up to 1.Wait, no, for x from 1/2 to 3/4, y ≥ x - 1/2 is a region that is above the line y = x - 1/2, which is from (1/2, 0) to (3/4, 1/4). So, the region is a trapezoid with vertices at (1/2, 0), (3/4, 1/4), (1, 1/2), and (1, 1). Wait, no, that's not right.Wait, I'm getting confused. Let me try to visualize it again.The line y = x - 1/2 starts at (1/2, 0) and goes up to (1, 1/2). So, above this line, for x from 1/2 to 1, y is from x - 1/2 to 1.So, the area above the line is a region bounded by x = 1/2, y = x - 1/2, y = 1, and x = 1.Wait, but for x from 1/2 to 3/4, y has to be ≥ x - 1/2, which is a line from (1/2, 0) to (3/4, 1/4). So, the region is a trapezoid with vertices at (1/2, 0), (3/4, 1/4), (1, 1/2), and (1, 1). Wait, but that's not a trapezoid because (1, 1/2) is connected to (1, 1), which is a vertical line.Wait, maybe it's a quadrilateral with vertices at (1/2, 0), (3/4, 1/4), (1, 1/2), and (1, 1). So, to find the area, I can split it into two parts: a triangle from (1/2, 0) to (3/4, 1/4) to (1, 1/2), and a triangle from (1, 1/2) to (1, 1) to (3/4, 1/4). Wait, no, that might not be accurate.Alternatively, I can use the shoelace formula for the quadrilateral.List the points in order:(1/2, 0), (3/4, 1/4), (1, 1/2), (1, 1), (1/2, 0).Compute the shoelace sum:Sum1 = (1/2)(1/4) + (3/4)(1/2) + (1)(1) + (1)(0) = (1/8) + (3/8) + 1 + 0 = 1/8 + 3/8 + 8/8 = 12/8 = 3/2.Sum2 = (0)(3/4) + (1/4)(1) + (1/2)(1) + (1)(1/2) = 0 + 1/4 + 1/2 + 1/2 = 1/4 + 1 = 5/4.Area = 1/2 |Sum1 - Sum2| = 1/2 |3/2 - 5/4| = 1/2 |6/4 - 5/4| = 1/2 * 1/4 = 1/8.Wait, so the area of the quadrilateral is 1/8.But earlier, I had two regions: one from x = 1/2 to 3/4 with area 1/32 and another from x = 3/4 to 1 with area 5/32, totaling 6/32 = 3/16.But according to the shoelace formula, the area above the line y = x - 1/2 is 1/8.So, which one is correct?Wait, 1/8 is 4/32, and 3/16 is 6/32. These are different.Wait, perhaps I'm misdefining the regions.Wait, the condition is that the slope is ≥ 1, which translates to y ≥ x - 1/2 for x ≥ 3/4 and y ≤ x - 1/2 for 1/2 ≤ x ≤ 3/4.So, the total area is the sum of the area where y ≥ x - 1/2 for x ≥ 3/4 and the area where y ≤ x - 1/2 for 1/2 ≤ x ≤ 3/4.So, the first area is the region above y = x - 1/2 from x = 3/4 to x = 1, which we calculated as 5/32.The second area is the region below y = x - 1/2 from x = 1/2 to x = 3/4, which we calculated as 1/32.So, total area is 5/32 + 1/32 = 6/32 = 3/16.But according to the shoelace formula, the area above y = x - 1/2 is 1/8, which is 4/32. So, why the discrepancy?Wait, perhaps the shoelace formula is including the region from x = 1/2 to x = 3/4 where y ≥ x - 1/2, but that region is actually split into two parts: one where y ≥ x - 1/2 (which is part of the total area) and one where y ≤ x - 1/2 (which is another part of the total area). So, the shoelace formula is calculating the entire area above y = x - 1/2 from x = 1/2 to x = 1, which is 1/8, but we only want the part where the slope is ≥ 1, which is the union of two regions: above the line for x ≥ 3/4 and below the line for 1/2 ≤ x ≤ 3/4.Wait, but the shoelace formula gave me 1/8 for the area above the line from x = 1/2 to x = 1, which is 1/8. But according to the integral, the area above the line for x ≥ 3/4 is 5/32, and the area below the line for 1/2 ≤ x ≤ 3/4 is 1/32, totaling 6/32 = 3/16.Wait, 3/16 is 6/32, which is less than 1/8 (which is 4/32). So, perhaps the shoelace formula is including more area than we need.Wait, no, the shoelace formula is calculating the area above the line y = x - 1/2 from x = 1/2 to x = 1, which is 1/8. But our condition is that the slope is ≥ 1, which is equivalent to y ≥ x - 1/2 for x ≥ 3/4 and y ≤ x - 1/2 for 1/2 ≤ x ≤ 3/4.So, the total area is the sum of these two regions: above the line for x ≥ 3/4 and below the line for 1/2 ≤ x ≤ 3/4.So, the area above the line for x ≥ 3/4 is 5/32, and the area below the line for 1/2 ≤ x ≤ 3/4 is 1/32, totaling 6/32 = 3/16.But the shoelace formula gave me 1/8 for the area above the line from x = 1/2 to x = 1, which is 4/32. So, why is there a difference?Wait, perhaps the shoelace formula is correct, and my integral is wrong.Wait, let me think again.The line y = x - 1/2 intersects the square at (1/2, 0) and (1, 1/2). So, the area above this line within the square is a quadrilateral with vertices at (1/2, 0), (1, 1/2), (1, 1), and (0, 1). Wait, no, because the line y = x - 1/2 doesn't intersect the top side at (0,1). It only intersects at (1, 1/2).Wait, actually, the area above the line y = x - 1/2 from x = 1/2 to x = 1 is a trapezoid with vertices at (1/2, 0), (1, 1/2), (1, 1), and (1/2, 1). Wait, no, because at x = 1/2, y = 0, and at x = 1, y = 1/2. So, the area above the line is bounded by x = 1/2, y = x - 1/2, y = 1, and x = 1.Wait, but that's not a trapezoid because the top boundary is y = 1, which is a horizontal line, and the bottom boundary is y = x - 1/2, which is a slant line.So, the area can be calculated as the integral from x = 1/2 to x = 1 of (1 - (x - 1/2)) dx = ∫ from 1/2 to 1 of (3/2 - x) dx.Compute this integral:Antiderivative is (3/2)x - (1/2)x².Evaluate at x = 1: (3/2)(1) - (1/2)(1) = 3/2 - 1/2 = 1.Evaluate at x = 1/2: (3/2)(1/2) - (1/2)(1/4) = 3/4 - 1/8 = 6/8 - 1/8 = 5/8.So, the area is 1 - 5/8 = 3/8.Wait, that's 3/8, which is 12/32. That's different from both previous results.Wait, now I'm really confused. Which one is correct?Wait, let me recast the integral.The area above the line y = x - 1/2 from x = 1/2 to x = 1 is the integral from x = 1/2 to x = 1 of (1 - (x - 1/2)) dx = ∫ from 1/2 to 1 of (3/2 - x) dx.Compute:Antiderivative is (3/2)x - (1/2)x².At x = 1: 3/2 - 1/2 = 1.At x = 1/2: (3/2)(1/2) - (1/2)(1/4) = 3/4 - 1/8 = 5/8.So, the area is 1 - 5/8 = 3/8.But according to the shoelace formula, the area of the quadrilateral (1/2, 0), (1, 1/2), (1, 1), (1/2, 1) is:Using shoelace:List the points: (1/2, 0), (1, 1/2), (1, 1), (1/2, 1), (1/2, 0).Compute Sum1:(1/2)(1/2) + (1)(1) + (1)(1) + (1/2)(0) = 1/4 + 1 + 1 + 0 = 2 + 1/4 = 9/4.Compute Sum2:(0)(1) + (1/2)(1) + (1)(1/2) + (1)(1/2) = 0 + 1/2 + 1/2 + 1/2 = 3/2.Area = 1/2 |9/4 - 3/2| = 1/2 |9/4 - 6/4| = 1/2 * 3/4 = 3/8.So, the area is indeed 3/8, which matches the integral.Wait, but earlier, when I considered the two regions separately, I got 3/16. So, where is the mistake?Ah, I see now. The condition is that the slope is ≥ 1, which translates to y ≥ x - 1/2 for x ≥ 3/4 and y ≤ x - 1/2 for 1/2 ≤ x ≤ 3/4.So, the total area is the sum of these two regions:1. For x ≥ 3/4, y ≥ x - 1/2: area = 5/32.2. For 1/2 ≤ x ≤ 3/4, y ≤ x - 1/2: area = 1/32.Total area = 5/32 + 1/32 = 6/32 = 3/16.But according to the shoelace formula, the area above the line y = x - 1/2 from x = 1/2 to x = 1 is 3/8, which is 12/32. So, why is there a discrepancy?Wait, because the condition is not just y ≥ x - 1/2, but also y ≤ x - 1/2 in a different region. So, the total area is not just the area above the line, but the union of two regions: above the line for x ≥ 3/4 and below the line for 1/2 ≤ x ≤ 3/4.So, the total area is 3/16, which is 6/32.But wait, 3/16 is less than 3/8, which makes sense because we're only considering part of the area above the line and part below.So, the probability is 3/16.But wait, earlier I thought it was 1/8, but that was incorrect.Wait, let me confirm.The total area where the slope is ≥ 1 is 3/16.So, the probability is 3/16, which simplifies to 3/16, so m = 3, n = 16, and m + n = 19.But wait, earlier I thought it was 1/8, but that was incorrect because I didn't account for both regions properly.Wait, let me double-check the calculations.For x from 3/4 to 1, y from x - 1/2 to 1:Area = ∫ from 3/4 to 1 of (1 - (x - 1/2)) dx = ∫ from 3/4 to 1 of (3/2 - x) dx.Antiderivative: (3/2)x - (1/2)x².At x = 1: 3/2 - 1/2 = 1.At x = 3/4: (3/2)(3/4) - (1/2)(9/16) = 9/8 - 9/32 = 27/32.So, area = 1 - 27/32 = 5/32.For x from 1/2 to 3/4, y from 0 to x - 1/2:Area = ∫ from 1/2 to 3/4 of (x - 1/2) dx.Antiderivative: (1/2)x² - (1/2)x.At x = 3/4: (1/2)(9/16) - (1/2)(3/4) = 9/32 - 3/8 = 9/32 - 12/32 = -3/32.At x = 1/2: (1/2)(1/4) - (1/2)(1/2) = 1/8 - 1/4 = -1/8.So, area = (-3/32) - (-1/8) = (-3/32) + 4/32 = 1/32.Total area = 5/32 + 1/32 = 6/32 = 3/16.Yes, that seems correct.So, the probability is 3/16, so m = 3, n = 16, and m + n = 19.Wait, but earlier I thought the shoelace formula gave me 1/8 for the area above the line, but that was for the entire area above the line from x = 1/2 to x = 1, which is 3/8. But our condition is only part of that area.So, the correct probability is 3/16.Wait, but let me think again. The slope condition is (y - 1/4)/(x - 3/4) ≥ 1.This can be rewritten as y - 1/4 ≥ x - 3/4 when x - 3/4 > 0, i.e., x > 3/4.And y - 1/4 ≤ x - 3/4 when x - 3/4 < 0, i.e., x < 3/4.But for x < 3/4, y ≤ x - 1/2, but y cannot be negative, so only for x ≥ 1/2, y can be ≤ x - 1/2.So, the regions are:1. x > 3/4: y ≥ x - 1/2.2. 1/2 ≤ x ≤ 3/4: y ≤ x - 1/2.So, the total area is the sum of these two regions, which we calculated as 3/16.Therefore, the probability is 3/16, so m + n = 3 + 16 = 19.Wait, but earlier I thought it was 1/8, but that was incorrect because I didn't account for both regions properly.So, the final answer is 19.

💡Okay, so I have this problem here about a line and an ellipse in the Cartesian coordinate plane. Let me try to figure it out step by step.First, part (1) asks for the parametric equation of line ( l ) and the Cartesian equation of ellipse ( C ).Starting with the line ( l ). It passes through the point ( P(1, 0) ) and has an angle of inclination of ( frac{pi}{3} ). I remember that the angle of inclination is the angle the line makes with the positive x-axis. So, the slope ( m ) of the line should be ( tanleft(frac{pi}{3}right) ). Calculating that, ( tanleft(frac{pi}{3}right) = sqrt{3} ). So, the slope is ( sqrt{3} ).Now, the parametric equations of a line can be written as:[begin{cases}x = x_0 + t cosalpha y = y_0 + t sinalphaend{cases}]where ( (x_0, y_0) ) is a point on the line, ( alpha ) is the angle of inclination, and ( t ) is the parameter.Given that the line passes through ( P(1, 0) ) and has an angle ( frac{pi}{3} ), substituting these values in, we get:[begin{cases}x = 1 + t cosleft(frac{pi}{3}right) y = 0 + t sinleft(frac{pi}{3}right)end{cases}]I know that ( cosleft(frac{pi}{3}right) = frac{1}{2} ) and ( sinleft(frac{pi}{3}right) = frac{sqrt{3}}{2} ). So, substituting these in, the parametric equations become:[begin{cases}x = 1 + frac{1}{2}t y = frac{sqrt{3}}{2}tend{cases}]Alright, that should be the parametric equation for line ( l ).Now, moving on to the ellipse ( C ). It has parametric equations:[begin{cases}x = costheta y = 2sinthetaend{cases}]I need to find its Cartesian equation. I remember that parametric equations of an ellipse can be converted to Cartesian form by eliminating the parameter ( theta ).From the parametric equations, I can express ( costheta = x ) and ( sintheta = frac{y}{2} ). Using the Pythagorean identity ( sin^2theta + cos^2theta = 1 ), substituting the expressions from above:[left(frac{y}{2}right)^2 + x^2 = 1]Simplifying that:[frac{y^2}{4} + x^2 = 1]So, the Cartesian equation of the ellipse is:[x^2 + frac{y^2}{4} = 1]That seems right. Let me double-check. If I plug in ( x = costheta ) and ( y = 2sintheta ) into the Cartesian equation, I should get:[cos^2theta + frac{(2sintheta)^2}{4} = cos^2theta + sin^2theta = 1]Yes, that works out. So, part (1) is done.Now, part (2) asks for the length of segment ( AB ) where line ( l ) intersects ellipse ( C ) at points ( A ) and ( B ).To find the points of intersection, I need to solve the system of equations consisting of the parametric equations of line ( l ) and the Cartesian equation of ellipse ( C ).Let me write down the parametric equations of line ( l ) again:[x = 1 + frac{1}{2}t y = frac{sqrt{3}}{2}t]I can substitute these into the ellipse equation ( x^2 + frac{y^2}{4} = 1 ).Substituting ( x ) and ( y ):[left(1 + frac{1}{2}tright)^2 + frac{left(frac{sqrt{3}}{2}tright)^2}{4} = 1]Let me compute each term step by step.First, expand ( left(1 + frac{1}{2}tright)^2 ):[1^2 + 2 times 1 times frac{1}{2}t + left(frac{1}{2}tright)^2 = 1 + t + frac{1}{4}t^2]Next, compute ( frac{left(frac{sqrt{3}}{2}tright)^2}{4} ):First, square ( frac{sqrt{3}}{2}t ):[left(frac{sqrt{3}}{2}tright)^2 = frac{3}{4}t^2]Then divide by 4:[frac{3}{4}t^2 times frac{1}{4} = frac{3}{16}t^2]Now, add both terms together:[1 + t + frac{1}{4}t^2 + frac{3}{16}t^2 = 1]Combine like terms:The ( t^2 ) terms: ( frac{1}{4}t^2 + frac{3}{16}t^2 = frac{4}{16}t^2 + frac{3}{16}t^2 = frac{7}{16}t^2 )The linear term: ( t )The constant term: ( 1 )So, the equation becomes:[frac{7}{16}t^2 + t + 1 = 1]Subtract 1 from both sides:[frac{7}{16}t^2 + t = 0]To make it easier, multiply both sides by 16 to eliminate the fraction:[7t^2 + 16t = 0]Factor out a ( t ):[t(7t + 16) = 0]So, the solutions are:[t = 0 quad text{or} quad 7t + 16 = 0 Rightarrow t = -frac{16}{7}]Therefore, the parameter ( t ) takes two values: ( t = 0 ) and ( t = -frac{16}{7} ).These correspond to the points ( A ) and ( B ) on the line ( l ). Let me find the coordinates of these points.For ( t = 0 ):[x = 1 + frac{1}{2}(0) = 1 y = frac{sqrt{3}}{2}(0) = 0]So, point ( A ) is ( (1, 0) ).For ( t = -frac{16}{7} ):[x = 1 + frac{1}{2}left(-frac{16}{7}right) = 1 - frac{8}{7} = frac{7}{7} - frac{8}{7} = -frac{1}{7} y = frac{sqrt{3}}{2}left(-frac{16}{7}right) = -frac{8sqrt{3}}{7}]So, point ( B ) is ( left(-frac{1}{7}, -frac{8sqrt{3}}{7}right) ).Now, I need to find the distance between points ( A(1, 0) ) and ( Bleft(-frac{1}{7}, -frac{8sqrt{3}}{7}right) ).The distance formula is:[AB = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}]Substituting the coordinates:[AB = sqrt{left(-frac{1}{7} - 1right)^2 + left(-frac{8sqrt{3}}{7} - 0right)^2}]Simplify each component:For the x-component:[-frac{1}{7} - 1 = -frac{1}{7} - frac{7}{7} = -frac{8}{7}]For the y-component:[-frac{8sqrt{3}}{7} - 0 = -frac{8sqrt{3}}{7}]Now, square both components:[left(-frac{8}{7}right)^2 = frac{64}{49} left(-frac{8sqrt{3}}{7}right)^2 = frac{64 times 3}{49} = frac{192}{49}]Add them together:[frac{64}{49} + frac{192}{49} = frac{256}{49}]Take the square root:[AB = sqrt{frac{256}{49}} = frac{16}{7}]So, the length of segment ( AB ) is ( frac{16}{7} ).Wait a second, let me verify if I did everything correctly. When I substituted ( t = -frac{16}{7} ) into the parametric equations, I got ( x = -frac{1}{7} ) and ( y = -frac{8sqrt{3}}{7} ). Then, using the distance formula between ( (1, 0) ) and ( left(-frac{1}{7}, -frac{8sqrt{3}}{7}right) ), I calculated the differences correctly.Calculating ( x_2 - x_1 = -frac{1}{7} - 1 = -frac{8}{7} ) and ( y_2 - y_1 = -frac{8sqrt{3}}{7} - 0 = -frac{8sqrt{3}}{7} ). Squaring both gives ( frac{64}{49} ) and ( frac{192}{49} ), which add up to ( frac{256}{49} ). Square root of that is ( frac{16}{7} ). That seems correct.Alternatively, I remember that in parametric equations, the distance between two points on a line can also be found using the parameter difference. The parametric equations use ( t ) as a parameter, and the distance between two points with parameters ( t_1 ) and ( t_2 ) is ( |t_2 - t_1| times sqrt{(cosalpha)^2 + (sinalpha)^2} ). Since ( sqrt{(cosalpha)^2 + (sinalpha)^2} = 1 ), the distance is just ( |t_2 - t_1| ).In this case, the two parameters are ( t = 0 ) and ( t = -frac{16}{7} ). So, the distance should be ( |0 - (-frac{16}{7})| = frac{16}{7} ). That matches my previous result.So, both methods give me the same answer, which is reassuring.Therefore, the length of segment ( AB ) is ( frac{16}{7} ).**Final Answer**The length of segment ( AB ) is boxed{dfrac{16}{7}}.

💡Okay, so I need to find the discriminant of this quadratic equation: 3x² + (3 + 1/3)x + 1/3. Hmm, I remember that the discriminant is part of the quadratic formula, right? It's the part under the square root, which is b² - 4ac. So, first, I should identify the coefficients a, b, and c from the equation.Let me write down the equation again to make sure I have it right: 3x² + (3 + 1/3)x + 1/3. So, comparing this to the standard quadratic form ax² + bx + c, I can see that a is 3, b is (3 + 1/3), and c is 1/3. Got that.Now, to find the discriminant, I need to calculate b squared minus four times a times c. Let me write that out: discriminant = b² - 4ac. So, substituting the values I have, that would be (3 + 1/3)² - 4 * 3 * (1/3).Wait, let me make sure I'm doing this correctly. First, I need to compute b squared. b is 3 + 1/3, so I need to square that. Squaring a binomial like (a + b)² is a² + 2ab + b². So, applying that here, (3 + 1/3)² would be 3² + 2 * 3 * (1/3) + (1/3)².Calculating each term: 3 squared is 9, 2 times 3 times 1/3 is... let me see, 2 * 3 is 6, and 6 * 1/3 is 2. Then, (1/3) squared is 1/9. So, adding those together: 9 + 2 + 1/9. That gives me 11 + 1/9, which is the same as 11.111... in decimal form, but I should keep it as a fraction for accuracy.Now, moving on to the second part of the discriminant: 4ac. Here, a is 3 and c is 1/3. So, 4 * 3 * (1/3). Let me compute that: 4 times 3 is 12, and 12 times 1/3 is 4. So, 4ac is 4.Putting it all together, the discriminant is b² - 4ac, which is (11 + 1/9) - 4. Subtracting 4 from 11 gives me 7, so now I have 7 + 1/9. To express this as a single fraction, 7 is the same as 63/9, so adding 1/9 gives me 64/9.Wait, let me double-check that. If I have 7 + 1/9, converting 7 to ninths: 7 * 9 is 63, so 63/9 + 1/9 is indeed 64/9. That seems right.So, the discriminant is 64/9. I think that's the answer. Let me just recap to make sure I didn't make any mistakes. I identified a, b, and c correctly. Then, I squared b by expanding the binomial, which gave me 9 + 2 + 1/9. Then, I calculated 4ac, which was 4. Subtracting 4 from 11 + 1/9 gave me 7 + 1/9, which is 64/9 when expressed as an improper fraction. Yeah, that all checks out.I don't think I made any calculation errors. The steps seem logical and straightforward. So, I feel confident that the discriminant is 64/9.

💡Okay, so I have this problem about two decks of playing cards. One deck is laid out on a table, and another deck is placed on top of each card. Some cards might match, and I need to find the expected number of matches and the variance of the number of matches. Hmm, let me think about how to approach this.First, for part (a), the expected number of matches. I remember that expectation can often be found by summing up the probabilities of each individual event. In this case, each pair of cards (one from the bottom deck and one from the top deck) could be a match. So maybe I can model this with indicator variables.Let me define an indicator variable ( I_k ) for each position ( k ) from 1 to ( N ) (assuming there are ( N ) cards in each deck). ( I_k ) will be 1 if the ( k )-th card from the bottom deck matches the ( k )-th card from the top deck, and 0 otherwise. Then, the total number of matches ( S ) would be the sum of all these ( I_k ) variables: ( S = I_1 + I_2 + dots + I_N ).Now, to find the expected value ( E[S] ), I can use the linearity of expectation. That means ( E[S] = E[I_1] + E[I_2] + dots + E[I_N] ). Each ( E[I_k] ) is just the probability that the ( k )-th cards match. Since the top deck is shuffled randomly, the probability that any specific card matches the bottom deck is ( frac{1}{N} ). So each ( E[I_k] = frac{1}{N} ).Adding them all up, ( E[S] = N times frac{1}{N} = 1 ). So the expected number of matches is 1. That seems straightforward.Now, moving on to part (b), the variance of the number of matches. Variance is a bit trickier because it involves the expectation of the square of the random variable minus the square of the expectation. So, ( text{Var}(S) = E[S^2] - (E[S])^2 ).I already know ( E[S] = 1 ), so ( (E[S])^2 = 1 ). Now I need to find ( E[S^2] ). Expanding ( S^2 ), we get ( S^2 = (I_1 + I_2 + dots + I_N)^2 ). This expands to the sum of each ( I_k^2 ) plus twice the sum of all ( I_j I_k ) for ( j neq k ).Since each ( I_k ) is an indicator variable, ( I_k^2 = I_k ). So, ( E[I_k^2] = E[I_k] = frac{1}{N} ). There are ( N ) such terms, so the sum of ( E[I_k^2] ) is ( N times frac{1}{N} = 1 ).Now, for the cross terms ( I_j I_k ) where ( j neq k ). ( I_j I_k ) is 1 only if both the ( j )-th and ( k )-th cards match. What's the probability of that happening? Well, if the top deck is shuffled uniformly at random, the probability that the ( j )-th card matches is ( frac{1}{N} ), and given that, the probability that the ( k )-th card also matches is ( frac{1}{N-1} ) because one card has already been matched. So, the joint probability is ( frac{1}{N} times frac{1}{N-1} = frac{1}{N(N-1)} ).There are ( N(N-1) ) such pairs ( (j, k) ) where ( j neq k ). So, the sum of ( E[I_j I_k] ) is ( N(N-1) times frac{1}{N(N-1)} = 1 ).Putting it all together, ( E[S^2] = 1 + 1 = 2 ). Therefore, the variance ( text{Var}(S) = E[S^2] - (E[S])^2 = 2 - 1 = 1 ).Wait, that seems too simple. Both the expectation and variance are 1? Let me double-check.For the expectation, yes, each position has a ( frac{1}{N} ) chance of matching, and there are ( N ) positions, so the expectation is 1. That makes sense.For the variance, the cross terms involve the covariance between different ( I_j ) and ( I_k ). Since the matches are not independent events, the covariance isn't zero. But in my calculation, I accounted for the joint probability correctly, and it turned out that the sum of all the cross terms also contributed 1 to ( E[S^2] ). So, subtracting 1 from 2 gives a variance of 1. Hmm, that seems consistent.I think my reasoning is correct. Both the expected number of matches and the variance are 1.**Final Answer**a) The expected number of matches is boxed{1}.b) The variance of the number of matches is boxed{1}.

💡Okay, so I have this equation here: (1/2)^18 multiplied by (1/81)^k equals y, and I'm told that k is 9. I need to find the value of y. Alright, let's break this down step by step.First, I need to understand what the equation is saying. It's telling me that if I take (1/2) raised to the 18th power and multiply it by (1/81) raised to the kth power, I'll get y. And since k is given as 9, I can substitute that into the equation.So, substituting k = 9, the equation becomes: (1/2)^18 * (1/81)^9 = y.Now, I need to simplify this expression to find y. Let's start by looking at each part separately. First, (1/2)^18. I know that raising a fraction to a power means raising both the numerator and the denominator to that power. So, (1/2)^18 is the same as 1^18 divided by 2^18. Since 1 raised to any power is still 1, this simplifies to 1 / 2^18.Next, let's look at (1/81)^9. Similarly, this is 1^9 divided by 81^9. Again, 1 raised to any power is 1, so this simplifies to 1 / 81^9.Now, I need to multiply these two simplified fractions together: (1 / 2^18) * (1 / 81^9). When multiplying fractions, I multiply the numerators together and the denominators together. So, the numerator becomes 1 * 1 = 1, and the denominator becomes 2^18 * 81^9.So, y = 1 / (2^18 * 81^9).Now, I need to see if I can simplify this further. Let's look at 81. I know that 81 is 9 squared, and 9 is 3 squared. So, 81 is 3^4. Therefore, 81^9 is (3^4)^9.Using the exponent rule (a^m)^n = a^(m*n), this becomes 3^(4*9) = 3^36.So, now, the denominator is 2^18 * 3^36.Therefore, y = 1 / (2^18 * 3^36).At this point, I can't simplify this any further without calculating the actual values of 2^18 and 3^36, which are quite large numbers. So, the simplified form of y is 1 divided by the product of 2 raised to the 18th power and 3 raised to the 36th power.I think that's as far as I can go in simplifying this expression. So, the value of y is 1 divided by (2^18 multiplied by 3^36).

💡Okay, so I have this problem about a shopping mall buying two types of water purifiers, A and B. I need to figure out the cost of each, the number of purchasing plans, and the total filters given away. Let me try to break this down step by step.Starting with part (1): What is the cost of each A and B water purifier?The problem says that each A water purifier costs 600 yuan less than each B. So, if I let the cost of B be x yuan, then A would be x - 600 yuan. It also mentions that the mall spends 36,000 yuan to buy A water purifiers, and this amount is twice the number of B water purifiers that can be bought with 27,000 yuan. Hmm, okay, so 36,000 yuan buys A purifiers, and 27,000 yuan buys B purifiers, but the number of A purifiers is twice the number of B purifiers.Let me write this as equations. Let’s denote the number of A purifiers as N_A and the number of B purifiers as N_B.So, N_A = 36,000 / (x - 600)And N_B = 27,000 / xAccording to the problem, N_A = 2 * N_B. So,36,000 / (x - 600) = 2 * (27,000 / x)Let me solve this equation for x.First, multiply both sides by (x - 600) and x to eliminate denominators:36,000 * x = 2 * 27,000 * (x - 600)Simplify:36,000x = 54,000(x - 600)Expand the right side:36,000x = 54,000x - 32,400,000Bring all terms to one side:36,000x - 54,000x = -32,400,000-18,000x = -32,400,000Divide both sides by -18,000:x = (-32,400,000) / (-18,000) = 1,800So, the cost of each B water purifier is 1,800 yuan, and each A is 1,800 - 600 = 1,200 yuan.Okay, that seems straightforward. Let me double-check:Number of A purifiers: 36,000 / 1,200 = 30Number of B purifiers: 27,000 / 1,800 = 15Is 30 twice 15? Yes, it is. So that checks out.Moving on to part (2): If the mall uses 60,000 yuan to purchase both A and B water purifiers, with the condition that the number of B water purifiers does not exceed 8, how many purchasing plans are there?Let’s denote the number of A purifiers as a and the number of B purifiers as b.We have the total cost equation:1,200a + 1,800b = 60,000And the constraint:b ≤ 8We need to find all possible integer values of a and b that satisfy these conditions.First, let's simplify the cost equation. Divide everything by 600:2a + 3b = 100So, 2a = 100 - 3bTherefore, a = (100 - 3b)/2Since a must be an integer, (100 - 3b) must be even. 3b must be even because 100 is even. Since 3 is odd, b must be even for 3b to be even.So, b must be even and b ≤ 8.Possible values for b: 0, 2, 4, 6, 8Let's compute a for each:- b = 0: a = (100 - 0)/2 = 50- b = 2: a = (100 - 6)/2 = 94/2 = 47- b = 4: a = (100 - 12)/2 = 88/2 = 44- b = 6: a = (100 - 18)/2 = 82/2 = 41- b = 8: a = (100 - 24)/2 = 76/2 = 38So, the possible pairs (a, b) are:(50, 0), (47, 2), (44, 4), (41, 6), (38, 8)But wait, the problem says "the number of B water purifiers purchased does not exceed 8", so b can be from 0 to 8, inclusive. But in our calculation, we have b = 0, 2, 4, 6, 8. So that's 5 purchasing plans.But let me check if b can be odd numbers as well. Wait, earlier I thought b must be even because 3b must be even for a to be integer. Let me verify:If b is odd, say b=1:a = (100 - 3*1)/2 = (100 - 3)/2 = 97/2 = 48.5, which is not integer.Similarly, b=3:a = (100 - 9)/2 = 91/2 = 45.5, not integer.So yes, only even values of b will give integer a. Therefore, the purchasing plans are 5.Wait, but in the initial problem statement, it says "the number of B water purifiers purchased does not exceed 8", so b can be 0,1,2,...,8. But since only even b gives integer a, we have 5 plans.But hold on, the problem says "the number of B water purifiers purchased does not exceed 8", but doesn't specify that b has to be at least 1. So, b=0 is allowed, meaning purchasing only A purifiers.So, the number of purchasing plans is 5.But let me think again. The problem says "purchasing both A and B water purifiers". Wait, does that mean they have to purchase at least one of each? If so, then b cannot be 0, and a cannot be 50. So, the purchasing plans would be (47,2), (44,4), (41,6), (38,8). So, 4 purchasing plans.Looking back at the problem statement: "If the shopping mall uses 60,000 yuan to purchase both A and B water purifiers, with the condition that the number of B water purifiers purchased does not exceed 8..."The phrase "purchase both A and B" suggests that they must buy at least one of each. So, b must be at least 1. But since b must be even, the smallest even number greater than or equal to 1 is 2. So, b=2,4,6,8.Therefore, the purchasing plans are 4.So, the answer is 4 purchasing plans.Moving on to part (3): Under the condition in (2), to promote sales, the mall introduces a promotion where customers can draw a prize for each water purifier purchased, and the winning customers will receive a filter for the same water purifier. The cost of each filter for A and B water purifiers is 400 yuan and 500 yuan, respectively. If all the water purifiers are sold out and the mall still makes a profit of 5,250 yuan after deducting the cost of the prizes, how many filters in total were given away for both types of water purifiers?Alright, so first, we need to figure out how many filters were given away for each type, such that the total profit after deducting the filter costs is 5,250 yuan.First, let's recall the selling prices: A is 1,350 yuan, B is 2,100 yuan.The cost prices are A:1,200, B:1,800.So, profit per A: 1,350 - 1,200 = 150 yuanProfit per B: 2,100 - 1,800 = 300 yuanTotal profit before considering filters is:For each purchasing plan (a, b):Total profit = 150a + 300bBut then, the mall gives away filters, which cost 400 yuan for A and 500 yuan for B. Let’s denote the number of filters given away for A as f_A and for B as f_B.So, total profit after filters = (150a + 300b) - (400f_A + 500f_B) = 5,250We need to find f_A + f_B.But we don't know which purchasing plan was used. So, we need to check each possible purchasing plan and see which one(s) can satisfy the profit condition.From part (2), the possible purchasing plans are:1. (47, 2)2. (44, 4)3. (41, 6)4. (38, 8)Let me compute the total profit before filters for each:1. (47, 2): 150*47 + 300*2 = 7,050 + 600 = 7,6502. (44, 4): 150*44 + 300*4 = 6,600 + 1,200 = 7,8003. (41, 6): 150*41 + 300*6 = 6,150 + 1,800 = 7,9504. (38, 8): 150*38 + 300*8 = 5,700 + 2,400 = 8,100So, the total profits before filters are 7,650; 7,800; 7,950; 8,100.After deducting the filter costs, the profit is 5,250. So, the filter costs must be:7,650 - 5,250 = 2,4007,800 - 5,250 = 2,5507,950 - 5,250 = 2,7008,100 - 5,250 = 2,850So, the total filter costs are 2,400; 2,550; 2,700; 2,850 respectively.We need to find non-negative integers f_A and f_B such that:400f_A + 500f_B = total filter costAnd f_A + f_B is the total number of filters.Let’s check each case:1. Total filter cost = 2,400We need to solve 400f_A + 500f_B = 2,400Divide both sides by 100: 4f_A + 5f_B = 24Looking for non-negative integers f_A, f_B.Let me express f_A in terms of f_B:4f_A = 24 - 5f_BSo, 24 - 5f_B must be divisible by 4.Let’s check possible f_B:f_B = 0: 24 divisible by 4? Yes, f_A=6f_B = 1: 24 -5=19, not divisible by 4f_B = 2: 24 -10=14, not divisible by 4f_B = 3: 24 -15=9, not divisible by 4f_B = 4: 24 -20=4, divisible by 4, f_A=1f_B = 5: 24 -25= -1, negative, stopSo, possible solutions:(f_A, f_B) = (6,0) or (1,4)Total filters: 6+0=6 or 1+4=5But the problem says "customers can draw a prize for each water purifier purchased", so the number of filters given away cannot exceed the number of water purifiers sold.In purchasing plan (47,2), total water purifiers sold = 47 + 2 = 49So, total filters given away cannot exceed 49. Both 6 and 5 are less than 49, so both are possible.But the problem doesn't specify any other constraints, so both are possible. However, the problem asks for the total number of filters given away, so it could be either 6 or 5.But let's check the other purchasing plans to see if they can also satisfy the profit condition.2. Total filter cost = 2,550Equation: 400f_A + 500f_B = 2,550Divide by 50: 8f_A + 10f_B = 51Hmm, 8f_A + 10f_B = 51Looking for integer solutions.But 8f_A + 10f_B must be even, since 8 and 10 are even, but 51 is odd. So, no solution.Therefore, this purchasing plan is invalid.3. Total filter cost = 2,700Equation: 400f_A + 500f_B = 2,700Divide by 100: 4f_A + 5f_B = 27Looking for non-negative integers f_A, f_B.Express f_A in terms of f_B:4f_A = 27 - 5f_B27 - 5f_B must be divisible by 4.Check f_B:f_B=0: 27, not divisible by 4f_B=1: 22, not divisible by 4f_B=2: 17, not divisible by 4f_B=3: 12, divisible by 4, f_A=3f_B=4: 7, not divisible by 4f_B=5: 2, not divisible by 4f_B=6: -3, negativeSo, only solution: (3,3)Total filters: 3+3=6Check against total water purifiers sold: purchasing plan (41,6), total sold=476 ≤47, so valid.4. Total filter cost = 2,850Equation: 400f_A + 500f_B = 2,850Divide by 50: 8f_A + 10f_B = 57Again, 8f_A +10f_B must be even, but 57 is odd. No solution.So, only purchasing plans (47,2) and (41,6) can satisfy the profit condition.For (47,2):Possible filters: 6 or 5For (41,6):Filters: 6But the problem says "the mall still makes a profit of 5,250 yuan after deducting the cost of the prizes". It doesn't specify which purchasing plan was used, so we need to consider both possibilities.But wait, the problem says "under the condition in (2)", which refers to the purchasing plans where b ≤8. So, both (47,2) and (41,6) are valid.But the problem asks for the total number of filters given away. Since both purchasing plans are possible, but the total filters could be 5, 6, or 6.Wait, but in purchasing plan (47,2), the total filters could be 5 or 6. In (41,6), it's 6.But the problem says "how many filters in total were given away for both types of water purifiers?" It doesn't specify which purchasing plan, so perhaps we need to consider all possibilities.But the problem also says "the mall still makes a profit of 5,250 yuan after deducting the cost of the prizes". So, it's referring to a specific scenario where the mall achieved this profit. Since both purchasing plans can achieve this profit with different numbers of filters, but the problem is asking for the total number of filters given away, it's ambiguous.Wait, maybe I need to see if both purchasing plans can lead to the same total filters. For (47,2), the total filters can be 5 or 6. For (41,6), it's 6.So, the possible total filters are 5 or 6.But the problem is asking for the answer directly, so maybe it's expecting a unique answer. Perhaps I need to check which purchasing plan is feasible.Wait, in purchasing plan (47,2), the number of B purifiers is 2, so the maximum number of filters for B is 2. But in the solution for (47,2), f_B can be 0 or 4. But f_B=4 would mean giving away 4 filters for B, but only 2 were sold. That's not possible.Ah, right! The number of filters given away cannot exceed the number of water purifiers sold for each type.So, in purchasing plan (47,2):f_A ≤47, f_B ≤2So, in the solutions:(f_A, f_B) = (6,0): okay(f_A, f_B) = (1,4): but f_B=4 >2, invalidSo, only (6,0) is valid. Therefore, total filters=6.Similarly, in purchasing plan (41,6):f_A ≤41, f_B ≤6Solution is (3,3): okay, total filters=6.So, both purchasing plans can only give away 6 filters each.Therefore, regardless of the purchasing plan, the total number of filters given away is 6.So, the answer is 6.**Final Answer**(1) The cost of each A water purifier is boxed{1200} yuan and each B water purifier is boxed{1800} yuan.(2) The shopping mall has boxed{4} purchasing plans.(3) The total number of filters given away is boxed{6}.