💡Okay, so I have this problem here about polar coordinates and converting them to rectangular coordinates. Let me try to understand what it's asking for.First, point P has polar coordinates (√3, π/2). I remember that in polar coordinates, the first number is the radius (ρ) and the second is the angle (θ). So, to convert this to rectangular coordinates, I think the formulas are x = ρ cos θ and y = ρ sin θ. Let me plug in the values:x = √3 * cos(π/2) y = √3 * sin(π/2)I know that cos(π/2) is 0 and sin(π/2) is 1, so that means x = 0 and y = √3. So, point P in rectangular coordinates is (0, √3). Got that down.Next, the curve C has the polar equation ρ = 4 cos(θ - π/3). I need to convert this to a rectangular equation. I remember that in general, ρ = a cos(θ - α) is a circle in polar coordinates. To convert it to rectangular, I can use the identity ρ = 4 cos(θ - π/3). Let me recall the formula for converting such equations.I think the formula is ρ = 2a cos(θ - α), which represents a circle with radius a centered at (a cos α, a sin α). Wait, in this case, it's 4 cos(θ - π/3), so maybe a is 2? Because 2a would be 4, so a = 2. Therefore, the center of the circle should be at (2 cos π/3, 2 sin π/3). Let me compute that:cos π/3 is 0.5, so 2 * 0.5 = 1 sin π/3 is (√3)/2, so 2 * (√3)/2 = √3So, the center is at (1, √3) and the radius is 2. Therefore, the rectangular equation should be (x - 1)^2 + (y - √3)^2 = 4. Let me verify that.Expanding the equation: (x - 1)^2 + (y - √3)^2 = 4 x^2 - 2x + 1 + y^2 - 2√3 y + 3 = 4 x^2 + y^2 - 2x - 2√3 y + 4 = 4 Simplify: x^2 + y^2 - 2x - 2√3 y = 0Hmm, that seems correct. So, curve C is a circle with center (1, √3) and radius 2.Now, part 1 is done: the parametric equation of line l and the rectangular equation of curve C. For line l, it has a slope of -1 and passes through point P (0, √3). So, the equation of the line in slope-intercept form is y = -x + √3. But they want the parametric equations.Parametric equations usually involve a parameter, say t, and express x and y in terms of t. Since the slope is -1, the direction vector can be (1, -1) or (-1, 1). Let me choose (1, -1) for simplicity. So, starting from point P (0, √3), the parametric equations would be:x = 0 + t y = √3 - tBut wait, sometimes people prefer to use a different parameter. Alternatively, using a direction vector scaled by √2 to make it a unit vector, but maybe that's complicating things. Let me stick with the simple one.So, parametric equations: x = t y = √3 - tBut let me check if that's correct. When t = 0, we are at (0, √3), which is point P. As t increases, x increases and y decreases, which makes sense for a slope of -1. Yeah, that seems right.Alternatively, sometimes people use a different parameter, like s, but it's the same idea. So, I think that's the parametric equation for line l.Moving on to part 2: If line l and curve C intersect at two points A and B, find the value of |PA| / |PB| + |PB| / |PA|.Hmm, okay. So, I need to find the points of intersection between line l and curve C, which are points A and B. Then, compute the distances from P to A and P to B, and then compute that expression.First, let's find the points of intersection. We have the parametric equations of line l:x = t y = √3 - tAnd the rectangular equation of curve C is (x - 1)^2 + (y - √3)^2 = 4.Let me substitute x and y from the parametric equations into the circle equation:(t - 1)^2 + (√3 - t - √3)^2 = 4 Simplify: (t - 1)^2 + (-t)^2 = 4 Expand both terms: (t^2 - 2t + 1) + t^2 = 4 Combine like terms: 2t^2 - 2t + 1 = 4 Subtract 4 from both sides: 2t^2 - 2t - 3 = 0So, the quadratic equation is 2t^2 - 2t - 3 = 0. Let me solve for t using the quadratic formula:t = [2 ± √(4 + 24)] / 4 Because discriminant D = b² - 4ac = 4 + 24 = 28 So, t = [2 ± √28] / 4 Simplify √28: √28 = 2√7 So, t = [2 ± 2√7] / 4 = [1 ± √7] / 2Therefore, the two values of t are (1 + √7)/2 and (1 - √7)/2.So, points A and B correspond to these t values. Let me denote t1 = (1 + √7)/2 and t2 = (1 - √7)/2.Now, I need to find |PA| and |PB|. Since P is the point (0, √3), and points A and B are on the line l, which is parametrized by t, I can think of |PA| and |PB| as the distances from P to A and P to B, which correspond to |t1| and |t2| multiplied by the scaling factor of the parameter t. Wait, actually, in parametric equations, the parameter t doesn't necessarily correspond directly to distance unless the direction vector is a unit vector.In my parametric equations, x = t and y = √3 - t, so the direction vector is (1, -1). The length of this vector is √(1^2 + (-1)^2) = √2. So, each unit of t corresponds to √2 units of distance.Therefore, the actual distance from P to A is |t1| * √2, and similarly for |PB|.But let me think again. Since the parametric equations are x = t and y = √3 - t, starting from P (0, √3), when t = 0, we are at P. As t increases, we move along the line. So, the parameter t here is not the actual distance, but it's scaled by the direction vector's length.So, to get the actual distance, we need to multiply t by the length of the direction vector, which is √2. Therefore, |PA| = |t1| * √2 and |PB| = |t2| * √2.But let me verify that. The distance between two points on a line given by parametric equations is |t2 - t1| times the length of the direction vector. Wait, actually, in this case, since we're measuring from P, which is at t = 0, the distance from P to A is |t1| * √2 and from P to B is |t2| * √2.But let me compute |PA| and |PB| directly using coordinates.Point A corresponds to t1 = (1 + √7)/2, so its coordinates are:x = t1 = (1 + √7)/2 y = √3 - t1 = √3 - (1 + √7)/2Similarly, point B corresponds to t2 = (1 - √7)/2, so its coordinates are:x = t2 = (1 - √7)/2 y = √3 - t2 = √3 - (1 - √7)/2Now, let's compute the distance from P (0, √3) to A and B.Distance formula: |PA| = √[(x_A - 0)^2 + (y_A - √3)^2] Similarly for |PB|.Compute |PA|:x_A = (1 + √7)/2 y_A = √3 - (1 + √7)/2So, y_A - √3 = - (1 + √7)/2Therefore, |PA| = √[( (1 + √7)/2 )^2 + ( - (1 + √7)/2 )^2 ] = √[ ( (1 + 2√7 + 7)/4 ) + ( (1 + 2√7 + 7)/4 ) ] = √[ (8 + 2√7)/4 + (8 + 2√7)/4 ] Wait, that doesn't seem right. Let me compute it step by step.First, ( (1 + √7)/2 )^2 = (1 + 2√7 + 7)/4 = (8 + 2√7)/4 = (4 + √7)/2Similarly, ( - (1 + √7)/2 )^2 is the same as above: (4 + √7)/2So, |PA| = √[ (4 + √7)/2 + (4 + √7)/2 ] = √[ (8 + 2√7)/2 ] = √[4 + √7]Similarly, compute |PB|:x_B = (1 - √7)/2 y_B = √3 - (1 - √7)/2So, y_B - √3 = - (1 - √7)/2Therefore, |PB| = √[ ( (1 - √7)/2 )^2 + ( - (1 - √7)/2 )^2 ] = √[ (1 - 2√7 + 7)/4 + (1 - 2√7 + 7)/4 ] = √[ (8 - 2√7)/4 + (8 - 2√7)/4 ] = √[ (16 - 4√7)/4 ] = √[4 - √7]Wait, that can't be right because 4 - √7 is positive, but let me verify.Wait, (1 - √7)/2 squared is (1 - 2√7 + 7)/4 = (8 - 2√7)/4 = (4 - √7)/2Similarly, the y-component squared is the same: (4 - √7)/2So, |PB| = √[ (4 - √7)/2 + (4 - √7)/2 ] = √[ (8 - 2√7)/2 ] = √[4 - √7]So, |PA| = √(4 + √7) and |PB| = √(4 - √7)Now, I need to compute |PA| / |PB| + |PB| / |PA|Let me denote |PA| = a and |PB| = bSo, the expression is a/b + b/a = (a² + b²) / (ab)Compute a² and b²:a² = 4 + √7 b² = 4 - √7So, a² + b² = (4 + √7) + (4 - √7) = 8ab = √(4 + √7) * √(4 - √7) = √[ (4 + √7)(4 - √7) ] = √[16 - (√7)^2] = √[16 - 7] = √9 = 3Therefore, the expression is 8 / 3So, the value of |PA| / |PB| + |PB| / |PA| is 8/3.Wait, but let me double-check my steps because sometimes I might have made a mistake in the algebra.First, when I computed |PA| and |PB|, I used the distance formula and got √(4 + √7) and √(4 - √7). Then, when I computed a² + b², it was 8, and ab was 3. So, 8/3 is the result.Alternatively, another approach is to use the properties of the quadratic equation. Since t1 and t2 are roots of 2t² - 2t - 3 = 0, then t1 + t2 = 1 and t1*t2 = -3/2But wait, in my earlier calculation, I had t1 = (1 + √7)/2 and t2 = (1 - √7)/2, so t1 + t2 = 1 and t1*t2 = (1 - 7)/4 = -6/4 = -3/2. So, that's correct.Now, the distances |PA| and |PB| are |t1| * √2 and |t2| * √2, but since t1 and t2 can be positive or negative, but in this case, t1 is positive and t2 is negative because √7 is about 2.645, so t1 ≈ (1 + 2.645)/2 ≈ 1.8225 and t2 ≈ (1 - 2.645)/2 ≈ -0.8225So, |t1| = t1 and |t2| = -t2Therefore, |PA| = t1 * √2 and |PB| = (-t2) * √2But let me see if I can express |PA| / |PB| + |PB| / |PA| in terms of t1 and t2.Let me denote |PA| = t1 * √2 and |PB| = (-t2) * √2Then, |PA| / |PB| = (t1 * √2) / (-t2 * √2) = -t1 / t2 Similarly, |PB| / |PA| = (-t2 * √2) / (t1 * √2) = -t2 / t1So, the expression becomes (-t1/t2) + (-t2/t1) = - (t1/t2 + t2/t1) = - ( (t1² + t2²) / (t1 t2) )But from earlier, t1 + t2 = 1 and t1 t2 = -3/2Compute t1² + t2² = (t1 + t2)² - 2 t1 t2 = 1² - 2*(-3/2) = 1 + 3 = 4So, t1² + t2² = 4Therefore, the expression is - (4 / (-3/2)) = - (4 * (-2/3)) = - (-8/3) = 8/3Wait, that's the same result as before. So, that confirms it.But in my earlier approach, I computed |PA| and |PB| directly and got 8/3. So, both methods lead to the same result.Therefore, the value is 8/3.I think that's solid. I don't see any mistakes in the reasoning now.

💡Okay, so Alice, Bob, and Carol are playing this game where each picks a number between 0 and 1, and the winner is the one whose number is in the middle. Alice picks her number uniformly at random from the entire interval [0,1], and Bob picks his number uniformly at random from [0.25, 0.75]. Carol gets to choose her number strategically to maximize her chance of being the winner. The options given are 0.25, 0.5, 0.625, and 0.75. I need to figure out which number Carol should pick.First, let me understand the problem better. The winner is the person whose number is between the other two. So, if Carol picks a number c, she wins if either Alice's number a is less than c and Bob's number b is greater than c, or if Alice's number a is greater than c and Bob's number b is less than c. So, Carol's probability of winning depends on the positions of a and b relative to c.Since Alice's number a is uniformly random over [0,1], the probability that a is less than c is just c, and the probability that a is greater than c is 1 - c. Bob's number b is uniformly random over [0.25, 0.75], so the probability that b is less than c depends on where c is relative to [0.25, 0.75]. Similarly, the probability that b is greater than c also depends on where c is.Let me break this down into cases based on where Carol picks c.**Case 1: c is less than 0.25**In this case, since Bob's number b is always at least 0.25, if Carol picks c < 0.25, then b will always be greater than c. So, for Carol to win, Alice's number a must be less than c. The probability of that happening is just c, since a is uniform over [0,1]. So, in this case, Carol's winning probability is c.**Case 2: c is between 0.25 and 0.75**Here, Carol's number c is within Bob's range. So, Bob's number b can be either less than c or greater than c. The probability that b is less than c is (c - 0.25)/(0.75 - 0.25) = (c - 0.25)/0.5 = 2(c - 0.25). Similarly, the probability that b is greater than c is (0.75 - c)/0.5 = 2(0.75 - c).Now, Carol can win in two ways:1. a < c and b > c2. a > c and b < cSo, the total probability is the sum of these two probabilities.First, the probability that a < c is c, and the probability that b > c is 2(0.75 - c). So, the probability for the first scenario is c * 2(0.75 - c).Second, the probability that a > c is (1 - c), and the probability that b < c is 2(c - 0.25). So, the probability for the second scenario is (1 - c) * 2(c - 0.25).Adding these together, Carol's winning probability in this case is:P(c) = 2c(0.75 - c) + 2(1 - c)(c - 0.25)Let me expand this:= 2c*0.75 - 2c² + 2(1 - c)(c - 0.25)= 1.5c - 2c² + 2[(1)(c - 0.25) - c(c - 0.25)]= 1.5c - 2c² + 2[c - 0.25 - c² + 0.25c]= 1.5c - 2c² + 2[1.25c - 0.25 - c²]= 1.5c - 2c² + 2.5c - 0.5 - 2c²= (1.5c + 2.5c) + (-2c² - 2c²) - 0.5= 4c - 4c² - 0.5Wait, that seems a bit complicated. Maybe I made a mistake in expanding. Let me try again.Starting from:P(c) = 2c(0.75 - c) + 2(1 - c)(c - 0.25)First term: 2c(0.75 - c) = 1.5c - 2c²Second term: 2(1 - c)(c - 0.25)Let me compute (1 - c)(c - 0.25):= 1*c - 1*0.25 - c*c + c*0.25= c - 0.25 - c² + 0.25c= (c + 0.25c) - 0.25 - c²= 1.25c - 0.25 - c²Multiply by 2:= 2.5c - 0.5 - 2c²Now, add the first term:1.5c - 2c² + 2.5c - 0.5 - 2c²= (1.5c + 2.5c) + (-2c² - 2c²) - 0.5= 4c - 4c² - 0.5Hmm, that seems correct. So, P(c) = -4c² + 4c - 0.5This is a quadratic function in terms of c. To find its maximum, since the coefficient of c² is negative, the parabola opens downward, so the maximum is at the vertex.The vertex occurs at c = -b/(2a) where the quadratic is in the form ax² + bx + c.Here, a = -4, b = 4.So, c = -4/(2*(-4)) = -4/(-8) = 0.5So, the maximum occurs at c = 0.5.Let me compute P(0.5):P(0.5) = -4*(0.5)^2 + 4*(0.5) - 0.5= -4*(0.25) + 2 - 0.5= -1 + 2 - 0.5= 0.5So, at c = 0.5, the probability is 0.5.But wait, let me check the endpoints of this interval.At c = 0.25:P(0.25) = -4*(0.25)^2 + 4*(0.25) - 0.5= -4*(0.0625) + 1 - 0.5= -0.25 + 1 - 0.5= 0.25At c = 0.75:P(0.75) = -4*(0.75)^2 + 4*(0.75) - 0.5= -4*(0.5625) + 3 - 0.5= -2.25 + 3 - 0.5= 0.25So, in this interval, the maximum probability is 0.5 at c = 0.5.**Case 3: c is greater than 0.75**In this case, since Bob's number b is always at most 0.75, if Carol picks c > 0.75, then b will always be less than c. So, for Carol to win, Alice's number a must be greater than c. The probability of that happening is 1 - c, since a is uniform over [0,1]. So, in this case, Carol's winning probability is 1 - c.Now, let's summarize:- For c < 0.25: P(c) = c- For 0.25 ≤ c ≤ 0.75: P(c) = -4c² + 4c - 0.5- For c > 0.75: P(c) = 1 - cNow, let's analyze these functions.In the first interval, P(c) increases from 0 to 0.25 as c increases from 0 to 0.25.In the second interval, we found that the maximum is at c = 0.5 with P(c) = 0.5.In the third interval, P(c) decreases from 0.25 to 0 as c increases from 0.75 to 1.So, comparing the maximums:- The maximum in the first interval is 0.25 at c = 0.25- The maximum in the second interval is 0.5 at c = 0.5- The maximum in the third interval is 0.25 at c = 0.75Therefore, the overall maximum probability is 0.5, achieved at c = 0.5.Wait, but the options given are 0.25, 0.5, 0.625, and 0.75. So, according to this, Carol should pick 0.5 to maximize her winning probability.But wait, let me double-check my calculations because sometimes when dealing with probabilities, especially with multiple variables, it's easy to make a mistake.Let me re-express the probability function for the second interval.P(c) = 2c(0.75 - c) + 2(1 - c)(c - 0.25)Let me compute this again step by step.First term: 2c(0.75 - c) = 1.5c - 2c²Second term: 2(1 - c)(c - 0.25)Let me compute (1 - c)(c - 0.25):= 1*c - 1*0.25 - c*c + c*0.25= c - 0.25 - c² + 0.25c= (c + 0.25c) - 0.25 - c²= 1.25c - 0.25 - c²Multiply by 2:= 2.5c - 0.5 - 2c²Now, add the first term:1.5c - 2c² + 2.5c - 0.5 - 2c²= (1.5c + 2.5c) + (-2c² - 2c²) - 0.5= 4c - 4c² - 0.5Yes, that's correct. So, P(c) = -4c² + 4c - 0.5Taking derivative to find maximum:dP/dc = -8c + 4Set to zero:-8c + 4 = 0-8c = -4c = (-4)/(-8) = 0.5So, yes, the maximum is at c = 0.5, and P(0.5) = 0.5.Therefore, Carol should pick 0.5 to maximize her winning probability.But wait, let me think again. Is there a possibility that choosing c = 0.625 could give a higher probability? Maybe I need to check the value of P(c) at c = 0.625.Let me compute P(0.625):P(0.625) = -4*(0.625)^2 + 4*(0.625) - 0.5= -4*(0.390625) + 2.5 - 0.5= -1.5625 + 2.5 - 0.5= -1.5625 + 2= 0.4375So, P(0.625) = 0.4375, which is less than 0.5.Similarly, at c = 0.75, P(c) = 0.25, which is less than 0.5.At c = 0.25, P(c) = 0.25, which is less than 0.5.Therefore, the maximum is indeed at c = 0.5 with P(c) = 0.5.So, the answer should be B) 0.5.

💡First, I need to determine the total number of carrots Carol and her mom initially picked. Carol picked 29 carrots, and her mom picked 16 carrots.Next, I'll add these two amounts together to find the total number of carrots they picked initially.After that, I know they had a total of 83 bad carrots. To find out how many more carrots they picked beyond the initial amount, I'll subtract the initial number of carrots from the total number of bad carrots.This will give me the number of additional carrots they picked.

💡Okay, so I have this problem where there are two circles of radius ( r ) that are externally tangent to each other and internally tangent to the ellipse given by the equation ( 4x^2 + 9y^2 = 18 ). I need to find the value of ( r ).First, I should probably sketch the situation to visualize it better. The ellipse equation is ( 4x^2 + 9y^2 = 18 ). To understand its shape, I can rewrite it in standard form by dividing both sides by 18:[frac{x^2}{frac{18}{4}} + frac{y^2}{frac{18}{9}} = 1]Simplifying that, I get:[frac{x^2}{4.5} + frac{y^2}{2} = 1]So, the ellipse has a semi-major axis of length ( sqrt{4.5} ) along the x-axis and a semi-minor axis of length ( sqrt{2} ) along the y-axis. Calculating those, ( sqrt{4.5} ) is approximately 2.121, and ( sqrt{2} ) is approximately 1.414.Now, the two circles are externally tangent to each other and internally tangent to this ellipse. Since they're externally tangent, the distance between their centers should be equal to the sum of their radii. But since both circles have the same radius ( r ), the distance between their centers is ( 2r ).Given the symmetry of the ellipse, it's logical to assume that the centers of the two circles lie along the major axis of the ellipse, which is the x-axis in this case. So, the centers of the circles should be at ( (h, 0) ) and ( (-h, 0) ) for some ( h ).Since the circles are internally tangent to the ellipse, each circle will touch the ellipse at exactly one point. Let's focus on one of the circles, say the one on the right at ( (h, 0) ). The equation of this circle is:[(x - h)^2 + y^2 = r^2]This circle must satisfy the ellipse equation as well because they are tangent. So, substituting ( y^2 ) from the circle's equation into the ellipse equation:From the circle:[y^2 = r^2 - (x - h)^2]Substitute into the ellipse:[4x^2 + 9(r^2 - (x - h)^2) = 18]Let me expand this:[4x^2 + 9r^2 - 9(x^2 - 2hx + h^2) = 18][4x^2 + 9r^2 - 9x^2 + 18hx - 9h^2 = 18]Combine like terms:[(4x^2 - 9x^2) + 18hx + (9r^2 - 9h^2) = 18][-5x^2 + 18hx + 9r^2 - 9h^2 = 18]Now, since the circle is tangent to the ellipse, this equation should have exactly one solution for ( x ). That means the quadratic equation in ( x ) must have a discriminant of zero.Let me rearrange the equation:[-5x^2 + 18hx + (9r^2 - 9h^2 - 18) = 0]Multiply both sides by -1 to make it a bit simpler:[5x^2 - 18hx + (-9r^2 + 9h^2 + 18) = 0]So, the quadratic equation is:[5x^2 - 18hx + (9h^2 - 9r^2 + 18) = 0]For this quadratic to have exactly one solution, the discriminant ( D ) must be zero.The discriminant ( D ) is given by:[D = b^2 - 4ac]Where ( a = 5 ), ( b = -18h ), and ( c = 9h^2 - 9r^2 + 18 ).Calculating ( D ):[D = (-18h)^2 - 4 times 5 times (9h^2 - 9r^2 + 18)][D = 324h^2 - 20(9h^2 - 9r^2 + 18)][D = 324h^2 - 180h^2 + 180r^2 - 360][D = (324h^2 - 180h^2) + 180r^2 - 360][D = 144h^2 + 180r^2 - 360]Set ( D = 0 ):[144h^2 + 180r^2 - 360 = 0]Divide the entire equation by 36 to simplify:[4h^2 + 5r^2 - 10 = 0][4h^2 + 5r^2 = 10]So, that's one equation relating ( h ) and ( r ).Now, I need another equation to solve for ( h ) and ( r ). Since the two circles are externally tangent to each other, the distance between their centers is ( 2h ), which should be equal to ( 2r ) because each has radius ( r ) and they are externally tangent.Wait, hold on. If the centers are at ( (h, 0) ) and ( (-h, 0) ), the distance between them is ( 2h ). Since the circles are externally tangent, the distance between centers should be equal to the sum of their radii, which is ( r + r = 2r ). So,[2h = 2r][h = r]So, ( h = r ). That's helpful. Now, substitute ( h = r ) into the equation we had earlier:[4h^2 + 5r^2 = 10][4r^2 + 5r^2 = 10][9r^2 = 10][r^2 = frac{10}{9}][r = sqrt{frac{10}{9}} = frac{sqrt{10}}{3}]Wait, but hold on a second. Let me verify this because I might have made a mistake.Earlier, I substituted ( h = r ) into the equation ( 4h^2 + 5r^2 = 10 ), which gives ( 9r^2 = 10 ), so ( r = sqrt{10}/3 ). But when I think about the ellipse, the semi-major axis is ( sqrt{4.5} approx 2.121 ), and the radius of the circle is ( sqrt{10}/3 approx 1.054 ). That seems plausible because the circle should fit inside the ellipse.But let me check my earlier steps again to make sure I didn't make a mistake.Starting from the discriminant:We had the quadratic equation in ( x ):[5x^2 - 18hx + (9h^2 - 9r^2 + 18) = 0]Discriminant:[D = (-18h)^2 - 4 times 5 times (9h^2 - 9r^2 + 18)][D = 324h^2 - 20(9h^2 - 9r^2 + 18)][D = 324h^2 - 180h^2 + 180r^2 - 360][D = 144h^2 + 180r^2 - 360]Set ( D = 0 ):[144h^2 + 180r^2 = 360]Divide by 36:[4h^2 + 5r^2 = 10]Yes, that's correct.Then, since the distance between centers is ( 2h ), and since the circles are externally tangent, ( 2h = 2r ), so ( h = r ). Substituting ( h = r ):[4r^2 + 5r^2 = 10][9r^2 = 10][r^2 = frac{10}{9}][r = frac{sqrt{10}}{3}]So, that seems consistent.But wait, the original ellipse equation was ( 4x^2 + 9y^2 = 18 ). Let me check if the circle with radius ( sqrt{10}/3 ) is indeed tangent to the ellipse.Take the circle on the right: ( (x - r)^2 + y^2 = r^2 ). Plugging ( r = sqrt{10}/3 ), the equation becomes:[(x - sqrt{10}/3)^2 + y^2 = (sqrt{10}/3)^2][x^2 - (2sqrt{10}/3)x + 10/9 + y^2 = 10/9][x^2 - (2sqrt{10}/3)x + y^2 = 0]Now, substitute ( y^2 ) from the circle into the ellipse equation:From the circle: ( y^2 = (2sqrt{10}/3)x - x^2 )Substitute into the ellipse:[4x^2 + 9[(2sqrt{10}/3)x - x^2] = 18][4x^2 + 6sqrt{10}x - 9x^2 = 18][-5x^2 + 6sqrt{10}x - 18 = 0]Multiply by -1:[5x^2 - 6sqrt{10}x + 18 = 0]Compute discriminant:[D = ( -6sqrt{10} )^2 - 4 times 5 times 18][D = 360 - 360 = 0]So, discriminant is zero, which confirms that the circle is tangent to the ellipse at exactly one point. Therefore, ( r = sqrt{10}/3 ) is correct.Wait, but in the initial problem statement, the ellipse is ( 4x^2 + 9y^2 = 18 ). Let me confirm if my calculations align with this.Yes, when I substituted ( y^2 ) from the circle into the ellipse, I arrived at a quadratic with discriminant zero, which is consistent with tangency.Therefore, I think my solution is correct. The radius ( r ) is ( sqrt{10}/3 ).But just to be thorough, let me consider the position of the circle. The center is at ( (r, 0) ), so the circle extends from ( 0 ) to ( 2r ) along the x-axis. The ellipse extends up to ( sqrt{4.5} approx 2.121 ) along the x-axis. Since ( 2r = 2sqrt{10}/3 approx 2.108 ), which is slightly less than ( sqrt{4.5} approx 2.121 ), so the circle just fits inside the ellipse, touching it at one point. That seems reasonable.Alternatively, if I consider the point of tangency, it should lie along the line connecting the centers of the ellipse and the circle. Since both centers are on the x-axis, the point of tangency should also lie on the x-axis. Let me check that.For the circle on the right, the point of tangency on the x-axis would be at ( (r + r, 0) = (2r, 0) ). Plugging into the ellipse equation:[4(2r)^2 + 9(0)^2 = 18][16r^2 = 18][r^2 = 18/16 = 9/8][r = 3/(2sqrt{2}) approx 1.0607]Wait, that's different from ( sqrt{10}/3 approx 1.054 ). Hmm, so which one is correct?Wait, perhaps my assumption that the point of tangency is on the x-axis is incorrect. Because if the circle is tangent to the ellipse, the point of tangency might not necessarily be on the x-axis. It could be somewhere else.Wait, but in the case of circles tangent to an ellipse, the point of tangency doesn't have to be on the major axis unless specified. So, my earlier approach where I substituted the circle equation into the ellipse and found the discriminant was correct because it accounts for the tangency condition regardless of the point's location.But then, why does plugging in ( y = 0 ) give a different result? Because if I plug ( y = 0 ) into both equations, I get two points on the x-axis where the circle intersects the ellipse. But if the circle is only tangent at one point, which might not be on the x-axis, then plugging ( y = 0 ) might not give the correct tangency condition.Therefore, my initial method using the discriminant is more accurate because it ensures that there's only one intersection point, which could be anywhere on the ellipse, not necessarily on the x-axis.So, going back, my solution ( r = sqrt{10}/3 ) is correct.But let me calculate both values numerically to see how different they are.( sqrt{10}/3 approx 3.162/3 approx 1.054 )( 3/(2sqrt{2}) approx 3/2.828 approx 1.0607 )These are very close but not the same. So, which one is correct?Wait, perhaps my assumption that the point of tangency is on the x-axis is wrong. Let me think.If the circle is tangent to the ellipse, the point of tangency must satisfy both equations and the gradients (slopes) of the ellipse and the circle must be equal at that point.So, let me try this approach.Let the point of tangency be ( (x, y) ). Then, it must satisfy both the ellipse equation and the circle equation, and the derivatives at that point must be equal.First, the ellipse equation:[4x^2 + 9y^2 = 18]Differentiate implicitly with respect to ( x ):[8x + 18y frac{dy}{dx} = 0][frac{dy}{dx} = -frac{8x}{18y} = -frac{4x}{9y}]Now, the circle equation:[(x - h)^2 + y^2 = r^2]Differentiate implicitly:[2(x - h) + 2y frac{dy}{dx} = 0][frac{dy}{dx} = -frac{(x - h)}{y}]At the point of tangency, the derivatives must be equal:[-frac{4x}{9y} = -frac{(x - h)}{y}]Simplify:[frac{4x}{9y} = frac{(x - h)}{y}]Multiply both sides by ( y ):[frac{4x}{9} = x - h]Multiply both sides by 9:[4x = 9x - 9h][-5x = -9h][x = frac{9h}{5}]So, the x-coordinate of the point of tangency is ( frac{9h}{5} ).Now, since ( h = r ), as established earlier, this becomes:[x = frac{9r}{5}]Now, substitute this back into the ellipse equation to find ( y ):From the ellipse:[4x^2 + 9y^2 = 18][4left(frac{9r}{5}right)^2 + 9y^2 = 18][4 times frac{81r^2}{25} + 9y^2 = 18][frac{324r^2}{25} + 9y^2 = 18]Multiply both sides by 25 to eliminate denominators:[324r^2 + 225y^2 = 450]Divide by 9:[36r^2 + 25y^2 = 50]Now, from the circle equation:[left(frac{9r}{5} - rright)^2 + y^2 = r^2]Simplify ( frac{9r}{5} - r = frac{4r}{5} ):[left(frac{4r}{5}right)^2 + y^2 = r^2][frac{16r^2}{25} + y^2 = r^2][y^2 = r^2 - frac{16r^2}{25} = frac{9r^2}{25}][y = pm frac{3r}{5}]Now, substitute ( y^2 = frac{9r^2}{25} ) into the equation we got from the ellipse:[36r^2 + 25 times frac{9r^2}{25} = 50]Simplify:[36r^2 + 9r^2 = 50][45r^2 = 50][r^2 = frac{50}{45} = frac{10}{9}][r = frac{sqrt{10}}{3}]So, this confirms the earlier result. Therefore, the radius ( r ) is ( sqrt{10}/3 ).Wait, but earlier when I plugged ( y = 0 ) into both equations, I got a different value for ( r ). That must mean that the point of tangency is not on the x-axis, which makes sense because if the circle is tangent to the ellipse, the point of tangency could be anywhere, not necessarily on the major axis.Therefore, my initial solution using the discriminant was correct, and the point of tangency is not on the x-axis, which is why plugging ( y = 0 ) gave a different result.So, to summarize, the radius ( r ) is ( sqrt{10}/3 ).But wait, let me check the initial problem statement again to make sure I didn't misinterpret anything.The problem says: "Two circles of radius ( r ) are externally tangent to each other and internally tangent to the ellipse ( 4x^2 + 9y^2 = 18 ). Find ( r )."Yes, that's what I solved. So, I think my answer is correct.Just to be absolutely thorough, let me compute the numerical value of ( sqrt{10}/3 ) and see if it makes sense.( sqrt{10} approx 3.1623 ), so ( sqrt{10}/3 approx 1.054 ).The semi-major axis of the ellipse is ( sqrt{4.5} approx 2.121 ), so the diameter is about 4.242. The two circles each have a radius of about 1.054, so the distance between their centers is ( 2 times 1.054 = 2.108 ), which is slightly less than the semi-major axis. That seems reasonable because the circles are inside the ellipse.If the distance between centers was equal to the semi-major axis, the circles would just touch the ends of the major axis, but since they are tangent to the ellipse somewhere inside, the distance between centers is a bit less.Therefore, I'm confident that ( r = sqrt{10}/3 ) is the correct answer.**Final Answer**The radius ( r ) is boxed{dfrac{sqrt{10}}{3}}.

💡Okay, so I've got this quadratic equation here: x squared plus 5x minus 1 equals zero. The question is asking for the sum of the two real roots, x1 and x2. Hmm, quadratic equations... I remember they have something to do with the coefficients and the roots. Let me think.First, I recall that for any quadratic equation in the form ax² + bx + c = 0, there are some relationships between the coefficients (a, b, c) and the roots (x1, x2). I think it's called Vieta's formulas. Yeah, that sounds right. Vieta's formulas relate the sum and product of the roots to the coefficients of the equation.So, specifically, I think the sum of the roots, x1 + x2, is equal to negative b over a. And the product of the roots, x1 times x2, is equal to c over a. Let me write that down to make sure I have it right:Sum of roots: x1 + x2 = -b/aProduct of roots: x1 * x2 = c/aOkay, so in my equation, x² + 5x - 1 = 0, the coefficients are a = 1, b = 5, and c = -1. Plugging those into the sum formula should give me the answer.So, x1 + x2 = -b/a = -5/1 = -5. That seems straightforward. But wait, let me double-check to make sure I didn't make a mistake. Sometimes signs can be tricky.Looking back at the equation: x² + 5x - 1 = 0. Yes, a is positive 1, b is positive 5, and c is negative 1. So, applying the formula, it's negative b over a, which is negative 5 over 1, which is indeed -5.Just to be thorough, maybe I should solve the equation and find the actual roots to see if their sum is indeed -5. Let's try that.The quadratic formula is x = [-b ± sqrt(b² - 4ac)] / (2a). Plugging in the values:x = [-5 ± sqrt(25 - 4*1*(-1))]/2x = [-5 ± sqrt(25 + 4)]/2x = [-5 ± sqrt(29)]/2So, the two roots are (-5 + sqrt(29))/2 and (-5 - sqrt(29))/2. Let's add them together:[(-5 + sqrt(29))/2] + [(-5 - sqrt(29))/2] = (-5 + sqrt(29) -5 - sqrt(29))/2 = (-10)/2 = -5.Yep, that checks out. The sum is definitely -5. I think I'm confident with that answer now.I guess another way to think about it is by expanding the factored form of the quadratic. If the roots are x1 and x2, then the equation can be written as (x - x1)(x - x2) = 0. Expanding that gives x² - (x1 + x2)x + x1x2 = 0. Comparing that to the original equation x² + 5x -1 = 0, we can see that:- (x1 + x2) = 5, so x1 + x2 = -5x1x2 = -1That's consistent with what I found earlier. So, multiple methods confirm that the sum of the roots is -5.I wonder why they specifically mentioned that the roots are real. Maybe because sometimes quadratics can have complex roots, but in this case, since the discriminant is positive (b² - 4ac = 25 + 4 = 29), the roots are indeed real and distinct. So, that's why they emphasized that x1 and x2 are real roots.But regardless of whether the roots are real or complex, Vieta's formulas still hold, right? The sum and product of the roots can be found using those formulas regardless of their nature. So, even if the roots were complex, x1 + x2 would still be -5. But in this case, they are real, so it's just an extra piece of information.I think I've covered all the bases here. I used Vieta's formula, checked by solving the quadratic, and even thought about the factored form to confirm. All methods lead to the same conclusion. So, I'm pretty sure the answer is -5.

💡First, I need to determine the most cost-effective way to purchase between 65 and 80 pounds of grass seed. The garden center offers three bag sizes: 5-pound, 10-pound, and 25-pound bags, each with different prices.To find the best value, I'll calculate the price per pound for each bag size:- 5-pound bag: 13.80 / 5 pounds = 2.76 per pound- 10-pound bag: 20.43 / 10 pounds = 2.043 per pound- 25-pound bag: 32.25 / 25 pounds = 1.29 per poundThe 25-pound bag offers the lowest price per pound, so I should prioritize buying as many of these as possible without exceeding 80 pounds.Calculating how many 25-pound bags fit within the 80-pound limit:80 pounds ÷ 25 pounds per bag = 3.2 bagsSince I can't purchase a fraction of a bag, I'll buy 3 bags of 25 pounds:3 bags × 25 pounds = 75 poundsThis gives me 75 pounds, which is within the desired range of 65 to 80 pounds. The cost for these 3 bags is:3 bags × 32.25 = 96.75To reach closer to 80 pounds, I could add a 5-pound bag:75 pounds + 5 pounds = 80 poundsThe cost for one additional 5-pound bag is 13.80, making the total cost:96.75 + 13.80 = 110.55However, since the requirement is to purchase at least 65 pounds, and 75 pounds already satisfies this, there's no need for the extra 5-pound bag.Therefore, the least possible cost to purchase at least 65 pounds but no more than 80 pounds of grass seed is 96.75.