answer：Okay, so I have this problem where I need to find the square of a complex number z, which is given as z = 2 - 3i. Hmm, complex numbers can be a bit tricky, but I think I remember some rules about squaring them. Let me try to recall.First, I know that squaring a binomial like (a + b)^2 follows the formula a^2 + 2ab + b^2. So, maybe I can apply that here. Let me write it out step by step.So, z^2 is (2 - 3i)^2. Expanding that, it should be (2)^2 + 2*(2)*(-3i) + (-3i)^2. Let me compute each part separately.First part: (2)^2 is straightforward, that's 4.Second part: 2*(2)*(-3i). Let's compute that. 2*2 is 4, and 4*(-3i) is -12i. So that part is -12i.Third part: (-3i)^2. Hmm, this might be where I can make a mistake. I remember that i squared is -1, so let's see. (-3i)^2 is (-3)^2 * (i)^2. (-3)^2 is 9, and i^2 is -1, so 9*(-1) is -9. So that part is -9.Now, putting it all together: 4 (from the first part) minus 12i (from the second part) minus 9 (from the third part). So, 4 - 9 is -5, and then we have -12i. So, combining those, the result should be -5 - 12i.Wait, let me double-check to make sure I didn't make any mistakes. Maybe I should multiply it out the long way to confirm.So, (2 - 3i)*(2 - 3i). Let's use the distributive property.First, multiply 2 by 2, which is 4.Then, 2 times -3i is -6i.Next, -3i times 2 is another -6i.Finally, -3i times -3i. That would be 9i^2, and since i^2 is -1, that's 9*(-1) = -9.Now, adding all those up: 4 - 6i -6i -9. Combine like terms: 4 -9 is -5, and -6i -6i is -12i. So, yeah, that gives me -5 -12i again. Okay, that matches my earlier result.I think that makes sense. So, squaring 2 - 3i gives me -5 -12i. I don't see any mistakes in my calculations, so I'm pretty confident that's the correct answer.Just to recap, the steps were:1. Recognize that squaring a binomial involves using the formula (a + b)^2 = a^2 + 2ab + b^2.2. Apply this formula to (2 - 3i)^2, breaking it down into three parts.3. Compute each part separately, remembering that i^2 = -1.4. Combine the results, carefully handling the signs.5. Double-check by expanding the multiplication step by step to ensure no mistakes were made.This process helped me arrive at the final answer without confusion. It's important to be careful with the signs, especially when dealing with negative numbers and imaginary units. Taking it step by step and verifying each part helps prevent errors.Another thing I could do to verify is plug the original complex number into a calculator or use another method to compute the square. But since I don't have a calculator handy, doing it manually twice seems sufficient. Both methods gave me the same result, so I feel good about it.In summary, squaring the complex number 2 - 3i involves expanding the expression, carefully computing each term, and combining like terms. The result is another complex number, -5 -12i.

answer：Okay, so I have this problem about a rectangle ABCD. The sides AB and CD are 3 units long, and the sides BC and DA are 8 units long. The rectangle is going to be rotated twice: first, 90 degrees clockwise around point D, and then another 90 degrees clockwise around the new position of point C after the first rotation. I need to find the total path length that point A travels during these two rotations.First, I should probably visualize the rectangle. Let me sketch it out mentally. Let's label the rectangle with points A, B, C, D in order, so that AB is 3 units, BC is 8 units, CD is 3 units, and DA is 8 units. So, it's a rectangle that's longer vertically than horizontally if I imagine it standing on side DA.Now, the first rotation is 90 degrees clockwise around point D. I need to figure out where point A moves during this rotation. Since it's a rotation around D, point D will stay fixed, and the rest of the rectangle will swing around D. A rotation of 90 degrees means that point A will move along a quarter-circle arc with radius equal to the distance from A to D.Wait, what's the distance from A to D? Since ABCD is a rectangle, AD is one of the longer sides, which is 8 units. But hold on, actually, in a rectangle, the diagonals are equal, but here, AD is a side, not a diagonal. So, the distance from A to D is 8 units. So, when we rotate point A 90 degrees around D, it will move along a quarter-circle with radius 8 units. The length of this arc is (1/4) of the circumference of a circle with radius 8. The circumference is 2πr, so 2π*8 = 16π. A quarter of that is 4π. So, point A moves 4π units during the first rotation.But wait, let me double-check that. The distance from A to D is indeed 8 units because AD is a side of the rectangle. So, yes, the radius is 8, and a quarter-circle would be 4π. That seems right.Now, after the first rotation, the rectangle has moved. Point C, which was originally at the end of the long side BC, has moved to a new position, let's call it C'. Now, the rectangle is going to be rotated another 90 degrees clockwise, but this time around point C'. I need to figure out the path that point A takes during this second rotation.To do this, I need to determine the new position of point A after the first rotation and then find the distance from this new position to point C', which will be the radius for the second rotation.Let me think about the coordinates to make this clearer. Maybe assigning coordinates to the rectangle will help. Let's place point D at the origin (0,0). Then, since DA is 8 units, point A would be at (0,8). Point B would be at (3,8), and point C would be at (3,0).After the first rotation of 90 degrees clockwise around D (0,0), point A (0,8) will move. A 90-degree clockwise rotation around the origin transforms a point (x,y) to (y, -x). So, applying that to point A (0,8), it becomes (8,0). So, point A moves from (0,8) to (8,0). The distance from D (0,0) to A (0,8) is 8 units, and after rotation, it's at (8,0), which is also 8 units from D. So, the radius is still 8 units, and the arc length is 4π as I calculated before.Now, after the first rotation, the rectangle has been moved. Let's figure out where point C is now. Originally, point C was at (3,0). After a 90-degree clockwise rotation around D (0,0), point C will move. Using the same rotation formula: (x,y) becomes (y, -x). So, (3,0) becomes (0, -3). Wait, that can't be right because point C is on the same side as D, so maybe I need to adjust my coordinate system.Wait, perhaps I made a mistake in assigning coordinates. Let me try again. If I place point D at (0,0), then since DA is 8 units, and AB is 3 units, point A would be at (0,8), point B at (3,8), and point C at (3,0). So, point C is at (3,0). After a 90-degree clockwise rotation around D (0,0), point C (3,0) becomes (0, -3). But that would place it below the origin, which might complicate things. Alternatively, maybe I should consider the rotation in a different way.Wait, perhaps it's better to think about the rectangle after the first rotation. When we rotate the rectangle 90 degrees clockwise around D, the side DC, which was originally along the x-axis from (0,0) to (3,0), will now be along the negative y-axis from (0,0) to (0,-3). Similarly, point A, which was at (0,8), will move to (8,0) after rotation. So, the new position of point C is at (0,-3).Wait, that makes more sense. So, after the first rotation, point C is at (0,-3). So, the new position of point C is (0,-3). Now, the second rotation is 90 degrees clockwise around this new point C' (0,-3). So, I need to find the new position of point A after the first rotation and then determine the distance from this new position to C' to find the radius for the second rotation.After the first rotation, point A was at (8,0). Now, we need to rotate this point 90 degrees clockwise around C' (0,-3). To do this, I can translate the coordinate system so that C' is at the origin, perform the rotation, and then translate back.So, the coordinates of point A relative to C' are (8 - 0, 0 - (-3)) = (8, 3). Now, applying a 90-degree clockwise rotation to (8,3) gives (3, -8). Then, translating back by adding (0,-3), the new coordinates of point A are (3, -8 -3) = (3, -11).Wait, that seems like a lot of movement. Let me verify. The distance from A' (8,0) to C' (0,-3) is sqrt((8-0)^2 + (0 - (-3))^2) = sqrt(64 + 9) = sqrt(73). So, the radius for the second rotation is sqrt(73). Therefore, the arc length for the second rotation is (1/4) of the circumference of a circle with radius sqrt(73). The circumference is 2π*sqrt(73), so a quarter of that is (π*sqrt(73))/2.Therefore, the total path length traveled by point A is the sum of the two arc lengths: 4π + (π*sqrt(73))/2. To combine these, I can write it as (4 + sqrt(73)/2)π.Wait, let me make sure I didn't mix up the order of rotations. The first rotation was around D, moving A along a quarter-circle of radius 8, giving 4π. The second rotation was around C', which is now at (0,-3), and the distance from A' (8,0) to C' is sqrt(73), so the second arc length is (π*sqrt(73))/2. Adding them together gives the total path length.Yes, that seems correct. So, the total distance is (4 + sqrt(73)/2)π, which corresponds to option C.

answer：First, I need to convert (379_{11}) to base 10. Breaking it down:- The digit '3' is in the (11^2) place, so it contributes (3 times 121 = 363).- The digit '7' is in the (11^1) place, contributing (7 times 11 = 77).- The digit '9' is in the (11^0) place, contributing (9 times 1 = 9).Adding these together: (363 + 77 + 9 = 449).Next, I'll convert (3A9_{12}) to base 10. Here, 'A' represents 10 in base 12:- The digit '3' is in the (12^2) place, contributing (3 times 144 = 432).- The digit 'A' (10) is in the (12^1) place, contributing (10 times 12 = 120).- The digit '9' is in the (12^0) place, contributing (9 times 1 = 9).Adding these together: (432 + 120 + 9 = 561).Finally, I'll add the two base 10 numbers: (449 + 561 = 1010).

answer：First, I'll define the variables: let ( C ) represent the number of chocolate chip cookie boxes and ( P ) represent the number of plain cookie boxes.From the problem, I know two key pieces of information:1. The total number of boxes sold is 1,585: [ C + P = 1,585 ]2. The total revenue from the sales is 1,586.75: [ 1.25C + 0.75P = 1,586.75 ]To simplify the calculations, I'll eliminate the decimals by multiplying the second equation by 100: [ 125C + 75P = 158,675 ]Next, I'll solve the first equation for ( C ): [ C = 1,585 - P ]Substituting this expression for ( C ) into the modified second equation: [ 125(1,585 - P) + 75P = 158,675 ]Expanding and simplifying: [ 198,125 - 125P + 75P = 158,675 ] [ 198,125 - 50P = 158,675 ]Solving for ( P ): [ 50P = 39,450 ] [ P = frac{39,450}{50} = 789 ]Therefore, the girl scout sold 789 boxes of plain cookies.

answer：Okay, so I'm trying to figure out if 12 pharmacies are enough to cover a city grid where there are 10 streets running north-south and 10 streets running east-west, forming a 9x9 grid of squares. Each intersection is a square, and there are 100 squares in total. The distance between adjacent squares is 1 km. A walking distance pharmacy means you can reach it within 3 km along the streets. The goal is to place pharmacies such that no matter which street you're on and which direction you're going, you'll get closer to a pharmacy as you walk.First, I need to visualize the grid. It's like a chessboard with 10 rows and 10 columns, making 100 intersections. Each intersection is 1 km apart from its neighbors. So, if I'm at any intersection, I can walk up to 3 km north, south, east, or west and still be within the walking distance of a pharmacy.Now, if I place a pharmacy at a particular intersection, it can cover all the intersections within a 3 km radius in all directions. That means, from the pharmacy's location, it can cover intersections from 3 blocks north to 3 blocks south and 3 blocks east to 3 blocks west. So, effectively, each pharmacy can cover a 7x7 grid of intersections (since 3 blocks in each direction plus the center block makes 7 blocks total in each direction).But wait, the grid is only 9x9, so placing a pharmacy near the edge would mean it doesn't cover as much area as one placed in the center. For example, a pharmacy at the northernmost row can't cover 3 blocks north because there are no blocks beyond that. So, edge pharmacies would have reduced coverage.Next, I need to calculate how many intersections each pharmacy can cover. In the center, a pharmacy covers 7x7 = 49 intersections. But on the edges, it covers less. For instance, a pharmacy at the corner would only cover 4x4 = 16 intersections because it can't go beyond the grid.However, the problem states that any person walking along any street, regardless of direction, should get closer to a pharmacy. This means that along every street, there should be a pharmacy within 3 km in both directions. So, I need to ensure that on every street, there's a pharmacy every 3 km or less.Since the grid is 9 blocks long in each direction, and each block is 1 km, I can divide the grid into segments of 3 km. That would mean placing a pharmacy every 3 blocks. So, in each direction, I would need pharmacies at positions 1, 4, 7, and 10. But wait, position 10 is the edge, so maybe only up to position 7 to cover the last segment.But this is just for one direction. Since the grid is two-dimensional, I need to cover both north-south and east-west directions. So, I might need to place pharmacies in a grid pattern that ensures coverage in both directions.If I place pharmacies every 3 blocks in both directions, starting from the top-left corner, the positions would be (1,1), (1,4), (1,7), (4,1), (4,4), (4,7), (7,1), (7,4), (7,7). That's 9 pharmacies. But wait, that's only covering the main grid points. What about the edges?Actually, if I place pharmacies at (1,1), (1,4), (1,7), (4,1), (4,4), (4,7), (7,1), (7,4), (7,7), (10,1), (10,4), (10,7), that's 12 pharmacies. This way, every street segment of 3 km has a pharmacy at each end, ensuring that anyone walking along the street will get closer to a pharmacy as they move.But I'm not sure if this is the most efficient way. Maybe there's a way to overlap the coverage areas so that fewer pharmacies are needed. For example, if a pharmacy covers a 7x7 area, maybe some of these areas can overlap, reducing the total number needed.However, considering the edges, it's tricky because the coverage near the edges is less. So, maybe 12 pharmacies are necessary to ensure that every street segment is covered within 3 km in both directions.Wait, let me think again. If I place pharmacies every 3 blocks in both directions, starting from the top-left, I get 12 pharmacies as mentioned. But is there a way to reduce this number by strategically placing some pharmacies to cover multiple directions more effectively?Alternatively, maybe I can use a checkerboard pattern or some other pattern that ensures coverage with fewer pharmacies. But I'm not sure if that would work because the requirement is that on every street, in both directions, you get closer to a pharmacy. So, it's not just about covering the grid points, but ensuring that along every street line, the pharmacies are spaced appropriately.I think the initial approach of placing pharmacies every 3 blocks in both directions, resulting in 12 pharmacies, is the way to go. It ensures that no matter which street you're on and which direction you're going, you'll get closer to a pharmacy within 3 km.But I'm still a bit uncertain. Maybe there's a more optimal arrangement. Let me try to visualize it again. If I place pharmacies at (1,1), (1,4), (1,7), (4,1), (4,4), (4,7), (7,1), (7,4), (7,7), (10,1), (10,4), (10,7), that's 12 pharmacies. Each of these covers a 7x7 area, but near the edges, the coverage is reduced. However, since the requirement is that on every street, you get closer to a pharmacy, maybe this arrangement works.Alternatively, if I place pharmacies at (2,2), (2,5), (2,8), (5,2), (5,5), (5,8), (8,2), (8,5), (8,8), that's 9 pharmacies. But then, the coverage near the edges might not be sufficient because the distance from the edge to the nearest pharmacy would be more than 3 km in some directions.So, maybe 12 pharmacies are necessary to ensure that every street segment is covered within 3 km in both directions. I think that's the answer.

answer：Okay, so I have this problem about systematic sampling. Let me try to understand it step by step. We have 500 products numbered from 001 to 500. A sample is drawn using systematic sampling. The smallest two numbers in the sample are 007 and 032. I need to find the largest number in the sample.First, I remember that systematic sampling involves selecting every k-th element from a population. The key here is to find the interval k. Once we have k, we can determine how many samples we have and then find the largest number.Given the smallest two numbers are 007 and 032. So, the difference between these two numbers should give me the interval k. Let me calculate that: 32 - 7 = 25. So, k is 25.Wait, is that right? If the first number is 007, then adding 25 each time should give the next numbers. So, 007, 032, 057, and so on. That seems to make sense.Now, how many samples are there? Since the population size is 500 and the interval is 25, the sample size should be 500 divided by 25, which is 20. So, there are 20 numbers in the sample.To find the largest number, I can use the formula for the nth term of an arithmetic sequence: a_n = a_1 + (n-1)*k. Here, a_1 is 7, k is 25, and n is 20.Let me plug in the numbers: a_20 = 7 + (20-1)*25 = 7 + 19*25. Calculating 19*25: 20*25 is 500, minus 25 is 475. So, 7 + 475 = 482.Therefore, the largest number in the sample should be 482.Wait, let me double-check. If I start at 7 and add 25 nineteen times, I should end up at 482. Let me count a few terms to make sure: 7, 32, 57, 82, 107, 132, 157, 182, 207, 232, 257, 282, 307, 332, 357, 382, 407, 432, 457, 482. Yep, that's 20 terms, and the last one is 482. That seems correct.I think I got it. The largest number in the sample is 482.