💡To determine the smallest positive integer ( n ) such that ( sqrt{96n} ) is an integer, we start by factorizing ( 96 ).First, we express ( 96 ) as:[96 = 16 times 6 = 4^2 times 6]This shows that ( 96 ) is composed of ( 4^2 ) and ( 6 ).Next, we consider the expression ( sqrt{96n} ):[sqrt{96n} = sqrt{4^2 times 6 times n}]For this square root to be an integer, the product ( 6 times n ) must form a perfect square. The prime factorization of ( 6 ) is ( 2 times 3 ). To make ( 6n ) a perfect square, ( n ) must provide the missing factors to pair with those in ( 6 ). Specifically, ( n ) must include at least one factor of ( 2 ) and one factor of ( 3 ).The smallest number that satisfies this condition is ( 6 ) itself, as ( 6 = 2 times 3 ).Therefore, the smallest value of the positive integer ( n ) for which ( sqrt{96n} ) is an integer is ( 6 ).

💡First, I need to determine the speed of the slower train. I'll denote the speed of the slower train as ( v ) km/h.Since the trains are moving in opposite directions, their relative speed is the sum of their individual speeds. Therefore, the relative speed is ( v + 90 ) km/h.The combined length of the two trains is ( 1.10 ) km + ( 0.9 ) km = ( 2.00 ) km.The time taken to cross each other is given as ( 47.99999999999999 ) seconds, which is approximately ( 48 ) seconds. I need to convert the speed from km/h to km/s to match the time unit.I know that ( 1 ) km/h is equivalent to ( frac{1}{3600} ) km/s. Therefore, the relative speed in km/s is ( (v + 90) times frac{1}{3600} ).Using the formula ( text{Distance} = text{Speed} times text{Time} ), I can set up the equation:[ 2.00 text{ km} = (v + 90) times frac{1}{3600} text{ km/s} times 48 text{ s} ]Simplifying the equation:[ 2.00 = (v + 90) times frac{48}{3600} ][ 2.00 = (v + 90) times frac{4}{300} ][ 2.00 = (v + 90) times frac{1}{75} ][ 2.00 times 75 = v + 90 ][ 150 = v + 90 ][ v = 150 - 90 ][ v = 60 ]Therefore, the speed of the slower train is ( 60 ) km/h.

💡First, I'll denote the total number of workers in the company as ( W ).Given that a third of the workers do not have a retirement plan, the number of workers without a retirement plan is ( frac{1}{3}W ).20% of the workers without a retirement plan are women, so the number of women without a retirement plan is ( 0.20 times frac{1}{3}W = frac{1}{15}W ).Since 40% of the workers with a retirement plan are men, this means 60% are women. The number of workers with a retirement plan is ( W - frac{1}{3}W = frac{2}{3}W ). Therefore, the number of women with a retirement plan is ( 0.60 times frac{2}{3}W = frac{2}{5}W ).The total number of women in the company is the sum of women with and without a retirement plan:[text{Total women} = frac{1}{15}W + frac{2}{5}W]Combining these fractions with a common denominator:[text{Total women} = frac{1}{15}W + frac{6}{15}W = frac{7}{15}W]Given that 40% of the workers with a retirement plan are men, the number of men with a retirement plan is ( 0.40 times frac{2}{3}W = frac{8}{30}W ). Since the total number of men is 128:[frac{8}{30}W = 128]Solving for ( W ):[W = 128 times frac{30}{8} = 128 times 3.75 = 480]Finally, calculating the total number of women:[text{Total women} = frac{7}{15} times 480 = 7 times 32 = 224]

💡First, I need to define the variables based on the problem statement. Let ( x ) represent the cost of purchasing one aerial drone, and ( y ) represent the cost of purchasing one programming robot.The problem states that the cost of purchasing 3 aerial drones is the same as purchasing 4 programming robots. This gives me the first equation:[3x = 4y]Next, it mentions that the total cost of purchasing 6 aerial drones and 9 programming robots is 76,500 yuan. This leads to the second equation:[6x + 9y = 76500]Now, I compare these equations with the provided options to find the correct system of equations.

💡Okay, so I have this function y = g(x) that's defined in three different segments. I need to figure out what the graph of y = (1/3)g(x) + 2 looks like. Hmm, let me break this down step by step.First, I remember that when you have a function like y = a*g(x) + b, it's a combination of vertical scaling and shifting. The 'a' factor here is 1/3, which means it's a vertical compression by a factor of 1/3. The '+2' at the end means I need to shift the entire graph up by 2 units. So, I should apply these transformations to each segment of g(x) individually.Let me start with the first segment of g(x): y = -x - 2 for -3 ≤ x ≤ 0. Applying the vertical compression by 1/3, I multiply the entire expression by 1/3. So, that becomes y = (1/3)*(-x - 2) = - (1/3)x - 2/3. Then, I add 2 to shift it up. So, y = - (1/3)x - 2/3 + 2. Simplifying that, -2/3 + 2 is the same as 4/3, so the equation becomes y = - (1/3)x + 4/3. Okay, so the first segment is now y = - (1/3)x + 4/3 for the same interval, -3 ≤ x ≤ 0. Let me check if that makes sense. If I plug in x = -3, y should be - (1/3)*(-3) + 4/3 = 1 + 4/3 = 7/3. Wait, but the original g(x) at x = -3 was y = -(-3) - 2 = 3 - 2 = 1. After scaling and shifting, it's 7/3, which is approximately 2.33, which is higher than the original 1. That seems correct because we shifted it up by 2 units. Similarly, at x = 0, y = - (1/3)*0 + 4/3 = 4/3, which is about 1.33. The original at x = 0 was y = -0 - 2 = -2, and after scaling and shifting, it's 4/3. That also makes sense because we scaled it down and then shifted it up.Moving on to the second segment of g(x): y = sqrt(4 - (x - 2)^2) - 2 for 0 < x ≤ 2. This looks like a semicircle equation. Let me recall that sqrt(r^2 - (x - h)^2) + k is the upper half of a circle with center (h, k) and radius r. In this case, it's sqrt(4 - (x - 2)^2) - 2, so the center is at (2, -2) and the radius is 2. But since it's the square root, it's the upper half of the circle. However, the original function is defined from x = 0 to x = 2, so it's only a quarter-circle in the upper half.Now, applying the vertical compression by 1/3, I get y = (1/3)*sqrt(4 - (x - 2)^2) - (2/3). Then, adding 2 for the vertical shift, it becomes y = (1/3)*sqrt(4 - (x - 2)^2) - (2/3) + 2. Simplifying that, -2/3 + 2 is 4/3, so the equation is y = (1/3)*sqrt(4 - (x - 2)^2) + 4/3.Let me think about what this does to the original semicircle. The vertical compression by 1/3 reduces the radius from 2 to 2/3. So, the new radius is 2/3. The center was originally at (2, -2), but after shifting up by 2 units, the center becomes (2, -2 + 2) = (2, 0). Wait, no, that's not quite right. The vertical shift is applied after the compression. So, the center's y-coordinate is affected by both the compression and the shift.Wait, let's clarify. The original center is at (2, -2). When we apply the vertical compression by 1/3, the center's y-coordinate becomes (1/3)*(-2) = -2/3. Then, adding 2 shifts it up by 2 units, so the new center is at (2, -2/3 + 2) = (2, 4/3). So, the center is now at (2, 4/3) and the radius is 2/3. That makes sense because the original radius was 2, and compressing it by 1/3 gives 2/3.So, the second segment is now a semicircle centered at (2, 4/3) with radius 2/3, defined for 0 < x ≤ 2. That seems correct.Now, the third segment of g(x): y = 2(x - 2) for 2 < x ≤ 3.Applying the vertical compression by 1/3, we get y = (1/3)*2(x - 2) = (2/3)(x - 2). Then, adding 2 for the vertical shift, it becomes y = (2/3)(x - 2) + 2. Let's simplify that: (2/3)x - 4/3 + 2. Converting 2 to thirds, that's 6/3, so -4/3 + 6/3 = 2/3. Therefore, the equation is y = (2/3)x + 2/3.Let me check the endpoints. At x = 2, the original g(x) was y = 2(0) = 0. After scaling and shifting, y = (2/3)*2 + 2/3 = 4/3 + 2/3 = 6/3 = 2. Wait, but the original at x = 2 was 0, and after scaling and shifting, it's 2. That seems correct because we scaled it up and then shifted it up. At x = 3, the original g(x) was y = 2(1) = 2. After scaling and shifting, y = (2/3)*3 + 2/3 = 2 + 2/3 = 8/3, which is approximately 2.666. That also makes sense.Wait, but hold on. The original third segment is defined for 2 < x ≤ 3, so at x = 2, it's not included in this segment. However, the second segment ends at x = 2, so we need to make sure that the point at x = 2 is correctly handled. The second segment ends at x = 2 with y = (1/3)*sqrt(4 - (2 - 2)^2) + 4/3 = (1/3)*sqrt(4) + 4/3 = (1/3)*2 + 4/3 = 2/3 + 4/3 = 6/3 = 2. So, at x = 2, both the second and third segments give y = 2, which is consistent.Similarly, at x = 3, the third segment gives y = (2/3)*3 + 2/3 = 2 + 2/3 = 8/3, which is approximately 2.666. That seems correct.So, putting it all together, the transformed function y = (1/3)g(x) + 2 consists of three segments:1. A line segment from x = -3 to x = 0 with the equation y = - (1/3)x + 4/3.2. A semicircular arc from x = 0 to x = 2 centered at (2, 4/3) with radius 2/3.3. A line segment from x = 2 to x = 3 with the equation y = (2/3)x + 2/3.Let me visualize this. The first segment is a straight line starting at (-3, 7/3) and ending at (0, 4/3). The second segment is a semicircle that starts at (0, 4/3) and ends at (2, 2). The third segment is a straight line starting at (2, 2) and ending at (3, 8/3).Wait, let me confirm the starting and ending points. For the first segment at x = -3, y = - (1/3)*(-3) + 4/3 = 1 + 4/3 = 7/3 ≈ 2.333. At x = 0, y = 4/3 ≈ 1.333. So, the line goes from (-3, 7/3) to (0, 4/3).For the second segment, at x = 0, y = (1/3)*sqrt(4 - (0 - 2)^2) + 4/3 = (1/3)*sqrt(4 - 4) + 4/3 = 0 + 4/3 = 4/3. At x = 2, y = (1/3)*sqrt(4 - 0) + 4/3 = (1/3)*2 + 4/3 = 2/3 + 4/3 = 6/3 = 2. So, the semicircle starts at (0, 4/3) and ends at (2, 2).For the third segment, at x = 2, y = (2/3)*2 + 2/3 = 4/3 + 2/3 = 6/3 = 2. At x = 3, y = (2/3)*3 + 2/3 = 2 + 2/3 = 8/3 ≈ 2.666. So, the line goes from (2, 2) to (3, 8/3).This seems consistent. The entire graph is transformed correctly with the vertical compression and shift.I think I've covered all the steps. I applied the transformations to each segment, checked the endpoints to ensure continuity, and visualized the resulting graph. It looks like everything lines up correctly.

💡First, I need to determine the aspect ratio of the original painting by comparing its width to its height.The original painting is 15 inches wide and 10 inches tall, giving an aspect ratio of 15:10, which simplifies to 3:2.Next, I'll set up a proportion to maintain this aspect ratio for the print. The print is 37.5 inches wide, and I need to find its height.Using the proportion:15 / 10 = 37.5 / Print heightSolving for the Print height:(15 * Print height) = (37.5 * 10)Print height = (375) / 15Print height = 25Therefore, the print is 25 inches tall.