💡Alright, so I've got this problem about a cone. Let me try to understand it step by step. The problem says that the length of the generatrix (which I think is the slant height) of a cone is 1, and the radius of the base is r. It also mentions that the area of the unfolded side surface of the cone is 1/3 of the area of the circle where the sector is located. I need to find the ratio l/r.First, I need to recall what the unfolded side surface of a cone looks like. I remember that when you unfold a cone, you get a sector of a circle. The radius of this sector is equal to the slant height of the cone, which is given as 1. So, the radius of the sector is 1.Now, the area of the sector is 1/3 of the area of the entire circle. The area of the entire circle with radius 1 is π*(1)^2 = π. So, the area of the sector is (1/3)*π.I also know that the area of a sector can be calculated using the formula (θ/2)*r^2, where θ is the central angle in radians and r is the radius of the sector. In this case, the radius r is 1, so the area of the sector is (θ/2)*(1)^2 = θ/2.Since the area of the sector is (1/3)*π, I can set up the equation θ/2 = (1/3)*π. Solving for θ, I get θ = (2/3)*π radians.Okay, so the central angle of the sector is (2/3)*π radians. Now, I need to relate this to the cone. The arc length of the sector is equal to the circumference of the base of the cone. The arc length of a sector is given by θ*r, where θ is the central angle and r is the radius of the sector. Here, θ is (2/3)*π and r is 1, so the arc length is (2/3)*π*1 = (2/3)*π.The circumference of the base of the cone is 2πr, where r is the radius of the base of the cone. So, 2πr = (2/3)*π. Solving for r, I divide both sides by π to get 2r = (2/3). Then, divide both sides by 2 to get r = (1/3).Wait, that doesn't seem right. If r is (1/3), then the ratio l/r would be 1/(1/3) = 3. But I need to double-check my steps to make sure I didn't make a mistake.Let me go back. The area of the sector is 1/3 of the area of the circle, which is correct. The area of the sector is θ/2, and that equals (1/3)*π. So, θ = (2/3)*π. That seems right.Then, the arc length of the sector is θ*r = (2/3)*π*1 = (2/3)*π. The circumference of the base of the cone is 2πr. Setting them equal: 2πr = (2/3)*π. Dividing both sides by π gives 2r = (2/3). Dividing both sides by 2 gives r = (1/3). So, l/r = 1/(1/3) = 3.Hmm, that seems consistent. Maybe I was overcomplicating it. So, the ratio of l to r is 3.But wait, let me think again. The generatrix is the slant height, which is 1, and the radius of the base is r. The arc length of the sector is equal to the circumference of the base of the cone. So, arc length = 2πr = θ*l, where θ is the central angle and l is the radius of the sector, which is the generatrix of the cone.Wait, I think I made a mistake earlier. The formula for arc length is θ*l, where l is the radius of the sector, which is the generatrix of the cone. So, arc length = θ*l = 2πr.Given that θ = (2/3)*π and l = 1, then θ*l = (2/3)*π*1 = (2/3)*π.So, 2πr = (2/3)*π. Dividing both sides by π gives 2r = (2/3). Then, r = (1/3).So, l/r = 1/(1/3) = 3.Okay, that seems correct. So, the ratio of l to r is 3.

💡First, I need to understand the function # p, which is defined as 2p - 20. This means that for any number p, the function multiplies it by 2 and then subtracts 20.The problem asks for the value of p such that applying the function three times in a row equals 6. In other words, I need to find p where #(#(#p)) = 6.To solve this, I'll start by applying the function once to p:#p = 2p - 20Next, I'll apply the function to the result of the first application:##p = #(#p) = 2(2p - 20) - 20 = 4p - 40 - 20 = 4p - 60Then, I'll apply the function to the result of the second application:###p = #(#(#p)) = 2(4p - 60) - 20 = 8p - 120 - 20 = 8p - 140Now, I know that ###p equals 6, so I can set up the equation:8p - 140 = 6To find p, I'll solve this equation:8p = 6 + 1408p = 146p = 146 / 8p = 18.25Therefore, the value of p is 18.25.

💡First, determine how many apples are in two dozen, which is (24) apples.Next, calculate the number of kunks needed for (24) apples. Since (3) kunks buy (5) apples, we set up a proportion:[3 text{ kunks} : 5 text{ apples} = x text{ kunks} : 24 text{ apples}][x = frac{3 times 24}{5} = frac{72}{5} = 14.4 text{ kunks}]Since we cannot have a fraction of a kunk in this context, we need to round up to (15) kunks to ensure we have enough apples.Next, calculate the number of lunks needed for (15) kunks. Since (7) lunks buy (4) kunks, we again set up a proportion:[7 text{ lunks} : 4 text{ kunks} = y text{ lunks} : 15 text{ kunks}][y = frac{7 times 15}{4} = frac{105}{4} = 26.25]Since we cannot have a fraction of a lunk in this context, we round up to (27) lunks to ensure we have enough kunks.Thus, to buy two dozen apples, (27) lunks are required.Conclusion:To purchase two dozen apples, (27) lunks are needed. Hence, the final answer is (boxed{27}) lunks.

💡Alright, so I have this problem here about two curves, C₁ and C₂. C₁ is given by the equation y = sin(x + π/2), and C₂ is y = sin(2x + 2π/3). The question is asking which of the four options (A, B, C, D) correctly describes how to transform C₁ into C₂.First, I need to understand what each transformation means. The options talk about stretching or shrinking the abscissa (which is the x-coordinate) and then shifting the curve either to the right or left. So, stretching the abscissa would mean making the graph wider, while shrinking it would make the graph narrower. Shifting to the right or left would involve moving the entire graph along the x-axis.Let me start by analyzing the equations of the two curves.For C₁: y = sin(x + π/2). I know that sin(x + π/2) is the same as cos(x), because shifting the sine function by π/2 to the left gives the cosine function. So, C₁ is essentially y = cos(x).For C₂: y = sin(2x + 2π/3). I can factor out the 2 inside the sine function: sin(2(x + π/3)). This tells me that the graph of C₂ is a sine function with a horizontal compression by a factor of 2 and a shift to the left by π/3 units.Now, I need to figure out how to transform C₁ into C₂. Since C₁ is y = cos(x), and C₂ is y = sin(2x + 2π/3), I need to see what operations will convert the cosine function into this transformed sine function.First, let's recall that cos(x) is the same as sin(x + π/2). So, starting from C₁: y = sin(x + π/2), I need to get to y = sin(2x + 2π/3).To do this, I think I need to perform two transformations: one that changes the period of the function (which would involve stretching or shrinking the abscissa) and one that shifts the function left or right.Looking at the argument of the sine function in C₂, it's 2x + 2π/3. In C₁, it's x + π/2. So, to go from x + π/2 to 2x + 2π/3, I need to somehow introduce the factor of 2 in front of x and adjust the phase shift accordingly.If I factor out the 2 from 2x + 2π/3, I get 2(x + π/3). So, this suggests that the function has been horizontally compressed by a factor of 2 and shifted to the left by π/3.But in C₁, the phase shift is π/2. So, how do I reconcile these two phase shifts?Let me think about the order of transformations. If I first compress the graph horizontally by a factor of 2, that would change x to 2x. Then, I need to adjust the phase shift accordingly.Starting with C₁: y = sin(x + π/2).If I horizontally compress by a factor of 2, replacing x with 2x, the equation becomes y = sin(2x + π/2).Now, I need to adjust this to match C₂: y = sin(2x + 2π/3).So, comparing y = sin(2x + π/2) with y = sin(2x + 2π/3), I need to adjust the phase shift from π/2 to 2π/3.To find out how much to shift, I can set up the equation:2x + π/2 + shift = 2x + 2π/3Subtracting 2x from both sides:π/2 + shift = 2π/3Solving for shift:shift = 2π/3 - π/2To subtract these fractions, I need a common denominator, which is 6.2π/3 = 4π/6π/2 = 3π/6So, shift = 4π/6 - 3π/6 = π/6Therefore, I need to shift the graph to the left by π/6 units.Wait, but in the options, the shifts are given as right or left shifts after stretching or shrinking. Let me make sure I'm interpreting this correctly.When I have y = sin(2x + π/2), and I want to get to y = sin(2x + 2π/3), I need to add an additional phase shift of π/6. Since the phase shift inside the function is added, it corresponds to a shift to the left by π/6.But let me double-check this. The general form is y = sin(Bx + C). The phase shift is -C/B. So, in y = sin(2x + π/2), the phase shift is -π/2 / 2 = -π/4, which is a shift to the left by π/4.In y = sin(2x + 2π/3), the phase shift is -2π/3 / 2 = -π/3, which is a shift to the left by π/3.So, the difference in phase shifts is π/3 - π/4 = π/12. So, to go from y = sin(2x + π/2) to y = sin(2x + 2π/3), I need to shift the graph to the left by an additional π/12.Wait, this contradicts my earlier conclusion. Which one is correct?Let me recast the problem. Starting from C₁: y = sin(x + π/2). I want to transform it into y = sin(2x + 2π/3).First, I can factor out the 2 from the argument of C₂: y = sin(2(x + π/3)).So, to get from y = sin(x + π/2) to y = sin(2(x + π/3)), I need to perform two transformations:1. Horizontal compression by a factor of 2, which replaces x with 2x, resulting in y = sin(2x + π/2).2. Then, adjust the phase shift from π/2 to 2π/3.Alternatively, I can think of it as first adjusting the phase shift and then compressing, but the order matters.Let me try both approaches.Approach 1:1. Start with y = sin(x + π/2).2. Horizontally compress by a factor of 2: y = sin(2x + π/2).3. Now, to adjust the phase shift from π/2 to 2π/3, I need to add an additional phase shift.The current phase shift after compression is -π/2 / 2 = -π/4 (shift left by π/4).The desired phase shift is -2π/3 / 2 = -π/3 (shift left by π/3).So, the additional shift needed is π/3 - π/4 = π/12 to the left.Therefore, the total transformation is:- Shrink the abscissa to half the original length (horizontal compression by factor 2).- Shift to the left by π/12.This matches option D.Approach 2:Alternatively, I could think of adjusting the phase shift first and then compressing.Starting with y = sin(x + π/2).If I want to shift it to the left by π/12, the new function would be y = sin(x + π/2 + π/12) = sin(x + 7π/12).Then, compressing horizontally by a factor of 2 would give y = sin(2x + 7π/12).But 7π/12 is not equal to 2π/3, which is 8π/12. So, this approach doesn't quite work.Therefore, the correct order is to first compress and then shift.So, the correct transformations are:- Shrink the abscissa to half the original length (horizontal compression by factor 2).- Shift to the left by π/12.Which corresponds to option D.I think my initial confusion was about whether to shift before or after compressing, but the correct order is to compress first and then shift, which leads to the conclusion that option D is correct.

💡Okay, so I have this problem here about a concert where three times as many children as adults attended. Adult tickets cost 7 and child tickets cost 3, and the total money collected was 6,000. I need to find out how many people bought tickets in total.Alright, let's break this down. First, I need to figure out how many adults and how many children attended. The problem says there are three times as many children as adults. So if I let the number of adults be some number, say, A, then the number of children would be 3 times that, which is 3A.Now, each adult ticket is 7, so the total money from adult tickets would be 7 times the number of adults, which is 7A. Similarly, each child ticket is 3, so the total money from child tickets would be 3 times the number of children, which is 3 times 3A, so that's 9A.Adding those two amounts together should give me the total money collected, which is 6,000. So, 7A plus 9A equals 6,000. That simplifies to 16A equals 6,000.To find A, I need to divide both sides by 16. So, A equals 6,000 divided by 16. Let me do that calculation. 6,000 divided by 16 is... hmm, 16 times 375 is 6,000 because 16 times 300 is 4,800 and 16 times 75 is 1,200, and 4,800 plus 1,200 is 6,000. So, A is 375.That means there are 375 adults. Since there are three times as many children, the number of children is 3 times 375, which is 1,125.To find the total number of people, I just add the number of adults and children together. So, 375 plus 1,125 equals 1,500.Wait, let me double-check that. If there are 375 adults at 7 each, that's 375 times 7, which is 2,625. And 1,125 children at 3 each is 1,125 times 3, which is 3,375. Adding those together, 2,625 plus 3,375 equals 6,000, which matches the total money collected. So, that seems correct.Therefore, the total number of people who bought tickets is 1,500.

💡First, I need to determine the percentage decrease in the length of the towel after bleaching. I'll start by defining the original length as ( L ) and the original breadth as ( B ). The original area of the towel is therefore ( A = L times B ).After bleaching, the length decreases by an unknown percentage ( x ), making the new length ( L' = L times (1 - frac{x}{100}) ). The breadth decreases by 10%, so the new breadth is ( B' = B times 0.9 ).The new area after bleaching is ( A' = L' times B' = L times (1 - frac{x}{100}) times B times 0.9 ).Given that the percentage decrease in area is approximately 28%, the new area is 72% of the original area, so ( A' = 0.72 times A ).Setting up the equation:[L times (1 - frac{x}{100}) times B times 0.9 = 0.72 times L times B]By canceling out ( L ) and ( B ) from both sides:[(1 - frac{x}{100}) times 0.9 = 0.72]Solving for ( x ):[1 - frac{x}{100} = frac{0.72}{0.9} = 0.8][frac{x}{100} = 1 - 0.8 = 0.2][x = 0.2 times 100 = 20]Therefore, the percentage decrease in length is 20%.