answer：Okay, so I have this problem where I'm given three vectors: OA, OB, and OC. Their coordinates are OA = (k, 12), OB = (4, 5), and OC = (10, 8). The question says that points A, B, and C are collinear, and I need to find the value of k. Hmm, collinear points mean that they all lie on the same straight line, right? So, I guess I need to use some vector properties or maybe coordinate geometry to figure this out.First, let me recall what it means for three points to be collinear. If points A, B, and C are collinear, then the vectors AB and AC should be scalar multiples of each other. Alternatively, the area of the triangle formed by these three points should be zero. I think either approach could work here. Maybe using vectors would be more straightforward since the problem gives vectors OA, OB, and OC.Let me write down the coordinates again to make sure I have them right:- OA = (k, 12)- OB = (4, 5)- OC = (10, 8)So, point A is at (k, 12), point B is at (4, 5), and point C is at (10, 8). I need to find k such that these three points lie on a straight line.One method I remember is that if three points are collinear, the slope between any two pairs of points should be equal. That is, the slope between A and B should be the same as the slope between B and C, or between A and C. Maybe I can use that.Let me calculate the slope between points B and C first. The slope formula is (y2 - y1)/(x2 - x1). So, for points B(4,5) and C(10,8), the slope would be (8 - 5)/(10 - 4) = 3/6 = 1/2. So, the slope between B and C is 1/2.Now, the slope between points A and B should also be 1/2 if they are collinear. Let me compute that. Point A is (k, 12) and point B is (4,5). So, the slope is (5 - 12)/(4 - k) = (-7)/(4 - k). This should equal 1/2.So, I can set up the equation:(-7)/(4 - k) = 1/2To solve for k, I can cross-multiply:-7 * 2 = 1 * (4 - k)Which simplifies to:-14 = 4 - kNow, subtract 4 from both sides:-14 - 4 = -k-18 = -kMultiply both sides by -1:18 = kSo, k is 18. Hmm, that seems straightforward. Let me verify it another way to make sure I didn't make a mistake.Another approach is to use vectors. If points A, B, and C are collinear, then the vector AB should be a scalar multiple of vector AC or BC. Let me try that.First, let's find vector AB. Vector AB is OB - OA, which is (4 - k, 5 - 12) = (4 - k, -7).Vector BC is OC - OB, which is (10 - 4, 8 - 5) = (6, 3).Since AB and BC should be parallel (because the points are collinear), vector AB must be a scalar multiple of vector BC. So, there exists some scalar λ such that:(4 - k, -7) = λ*(6, 3)This gives us two equations:1. 4 - k = 6λ2. -7 = 3λFrom the second equation, we can solve for λ:-7 = 3λ => λ = -7/3Now, plug this value of λ into the first equation:4 - k = 6*(-7/3) = -14So,4 - k = -14Subtract 4 from both sides:-k = -14 - 4 = -18Multiply both sides by -1:k = 18Same result as before. That's reassuring. Let me try another method just to be thorough.Another way is to use the concept that the area of the triangle formed by three collinear points is zero. The area can be calculated using the determinant formula:Area = (1/2)| (x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)) |If the area is zero, then the points are collinear.Plugging in the coordinates:x_A = k, y_A = 12x_B = 4, y_B = 5x_C = 10, y_C = 8So, the area is:(1/2)| k*(5 - 8) + 4*(8 - 12) + 10*(12 - 5) |Simplify each term:k*(-3) + 4*(-4) + 10*(7)Which is:-3k -16 + 70Combine constants:-3k + 54Set the area equal to zero:(1/2)| -3k + 54 | = 0Multiply both sides by 2:| -3k + 54 | = 0Which implies:-3k + 54 = 0Solving for k:-3k = -54k = (-54)/(-3) = 18Again, same result. So, k is definitely 18.Wait, just to make sure I didn't make any calculation errors, let me go through each method again briefly.First method: Slope between B and C is 1/2. Slope between A and B is (-7)/(4 - k). Setting them equal gives (-7)/(4 - k) = 1/2. Cross-multiplying: -14 = 4 - k => k = 18. Correct.Second method: Vectors AB and BC must be scalar multiples. AB = (4 - k, -7), BC = (6, 3). So, (4 - k)/6 = (-7)/3. Cross-multiplying: 3*(4 - k) = -42 => 12 - 3k = -42 => -3k = -54 => k = 18. Correct.Third method: Area determinant. Plugging in, got -3k + 54 = 0 => k = 18. Correct.All three methods lead to the same answer, so I can be confident that k is indeed 18.I think that's thorough enough. I don't see any mistakes in my reasoning, and all approaches confirm that k is 18.**Final Answer**The value of ( k ) is boxed{18}.

answer：Okay, so I have this problem where I need to find the number of ways to tile a 9x1 board using tiles of any length from 1x1 up to 9x1. Each tile can be colored either red, blue, or green. The catch is that I have to use all three colors at least once in the tiling. Then, I need to find the remainder when this number is divided by 1000.First, I need to figure out how many ways there are to tile the board without worrying about the colors. Since the board is 9 units long, each tiling corresponds to a composition of 9, where each part represents the length of a tile. The number of ways to partition the board into k tiles is given by the binomial coefficient C(8, k-1). For example, if I want to split the board into 3 tiles, the number of ways is C(8,2) which is 28.So, for each possible number of tiles (from 1 to 9), I can calculate the number of ways to partition the board. Then, for each partition, I need to consider the colorings. Since each tile can be one of three colors, if there are k tiles, there are 3^k colorings. However, this includes colorings where one or more colors might not be used. I need to subtract those cases where only one or two colors are used.This sounds like a classic inclusion-exclusion problem. The formula for the number of colorings using all three colors is 3^k - 3*2^k + 3. Here, 3^k is the total number of colorings, 3*2^k subtracts the cases where one color is missing, and then we add back 3 because we subtracted too much when considering two colors missing.So, for each k from 1 to 9, I need to compute the number of partitions (which is C(8, k-1)), multiply it by the number of valid colorings (which is 3^k - 3*2^k + 3), and then sum all these up to get the total number of tilings N.Let me list out the values for k from 1 to 9:- For k=1: C(8,0)=1, colorings=3^1 - 3*2^1 + 3 = 3 - 6 + 3 = 0. Hmm, that makes sense because if there's only one tile, you can't use all three colors.- For k=2: C(8,1)=8, colorings=3^2 - 3*2^2 + 3 = 9 - 12 + 3 = 0. Similarly, with two tiles, you can't use all three colors.- For k=3: C(8,2)=28, colorings=3^3 - 3*2^3 + 3 = 27 - 24 + 3 = 6.- For k=4: C(8,3)=56, colorings=3^4 - 3*2^4 + 3 = 81 - 48 + 3 = 36.- For k=5: C(8,4)=70, colorings=3^5 - 3*2^5 + 3 = 243 - 96 + 3 = 150.- For k=6: C(8,5)=56, colorings=3^6 - 3*2^6 + 3 = 729 - 192 + 3 = 540.- For k=7: C(8,6)=28, colorings=3^7 - 3*2^7 + 3 = 2187 - 384 + 3 = 1806.- For k=8: C(8,7)=8, colorings=3^8 - 3*2^8 + 3 = 6561 - 768 + 3 = 5610 + 3 = 5613? Wait, 6561 - 768 is 5793, plus 3 is 5796. Hmm, maybe I miscalculated earlier.- For k=9: C(8,8)=1, colorings=3^9 - 3*2^9 + 3 = 19683 - 1536 + 3 = 18150.Wait, let me double-check the calculations for k=8 and k=9 because I think I might have made a mistake.For k=8:3^8 = 65612^8 = 256So, 3*256 = 768Thus, 6561 - 768 + 3 = 6561 - 768 is 5793, plus 3 is 5796.For k=9:3^9 = 196832^9 = 5123*512 = 1536So, 19683 - 1536 + 3 = 19683 - 1536 is 18147, plus 3 is 18150.Okay, so I had a miscalculation earlier for k=8. It's 5796, not 5610. Similarly, for k=9, it's 18150.Now, let me list all the terms:- k=3: 28 * 6 = 168- k=4: 56 * 36 = 2016- k=5: 70 * 150 = 10500- k=6: 56 * 540 = 30240- k=7: 28 * 1806 = 50568- k=8: 8 * 5796 = 46368- k=9: 1 * 18150 = 18150Now, I need to sum all these up:168 + 2016 = 21842184 + 10500 = 1268412684 + 30240 = 4292442924 + 50568 = 9349293492 + 46368 = 139860139860 + 18150 = 158,010Wait, that doesn't match the initial answer of 167,480. Did I make a mistake in the multiplication?Let me check each multiplication:- k=3: 28 * 6 = 168 (correct)- k=4: 56 * 36: 56*30=1680, 56*6=336; 1680+336=2016 (correct)- k=5: 70 * 150: 70*100=7000, 70*50=3500; 7000+3500=10500 (correct)- k=6: 56 * 540: 56*500=28,000; 56*40=2,240; total 30,240 (correct)- k=7: 28 * 1806: Let's compute 28*1800=50,400 and 28*6=168; total 50,568 (correct)- k=8: 8 * 5796: 8*5000=40,000; 8*796=6,368; total 46,368 (correct)- k=9: 1 * 18,150 = 18,150 (correct)Now, adding them up step by step:Start with 168.168 + 2016 = 21842184 + 10,500 = 12,68412,684 + 30,240 = 42,92442,924 + 50,568 = 93,49293,492 + 46,368 = 139,860139,860 + 18,150 = 158,010Hmm, so according to my calculations, N is 158,010. But the initial answer was 167,480. There's a discrepancy here. Let me check if I made a mistake in the colorings for each k.Wait, for k=8, I had 3^8 - 3*2^8 + 3 = 6561 - 768 + 3 = 5796. But earlier, I thought it was 5610. So, 5796 is correct.Similarly, for k=9: 19683 - 1536 + 3 = 18150. Correct.Wait, maybe I missed some k values? The initial answer included k=3 to k=9, which is correct. But in my calculation, I got 158,010, but the initial answer was 167,480. So, I must have made a mistake somewhere.Wait, let me recalculate the sum:168 (k=3)+2016 (k=4) = 2184+10,500 (k=5) = 12,684+30,240 (k=6) = 42,924+50,568 (k=7) = 93,492+46,368 (k=8) = 139,860+18,150 (k=9) = 158,010Hmm, that's consistent. But the initial answer was 167,480. So, perhaps the initial answer had a mistake in the colorings for k=8 and k=9.Wait, in the initial answer, for k=8, it was 5610, but correct is 5796. Similarly, for k=9, it was 17,442, but correct is 18,150.So, let's recalculate N with the correct colorings:k=3: 28*6=168k=4:56*36=2016k=5:70*150=10,500k=6:56*540=30,240k=7:28*1806=50,568k=8:8*5796=46,368k=9:1*18,150=18,150Now, sum them up:168 + 2016 = 21842184 + 10,500 = 12,68412,684 + 30,240 = 42,92442,924 + 50,568 = 93,49293,492 + 46,368 = 139,860139,860 + 18,150 = 158,010So, N=158,010. Now, 158,010 divided by 1000 is 158 with a remainder of 10. So, the remainder is 10.But wait, the initial answer was 167,480, which modulo 1000 is 480. But according to my corrected calculations, it's 158,010, which modulo 1000 is 10.This suggests that there was a mistake in the initial colorings for k=8 and k=9. So, the correct N is 158,010, and the remainder modulo 1000 is 10.Wait, but let me double-check the colorings again.For k=8:3^8 = 65612^8 = 256So, 3*256=768Thus, 6561 - 768 + 3 = 6561 - 768 = 5793 +3=5796. Correct.For k=9:3^9=196832^9=5123*512=153619683 - 1536 +3=19683-1536=18147+3=18150. Correct.So, my calculations seem correct. Therefore, the initial answer might have had an error in the colorings for k=8 and k=9, leading to a higher N.Therefore, the correct N is 158,010, and 158,010 mod 1000 is 10.But wait, let me check if I missed any k values. The problem states that each tile can be any length from 1 to 9, so k can be from 1 to 9. However, for k=1 and k=2, the number of colorings using all three colors is zero, as we saw earlier. So, those terms are zero and don't contribute to N.Therefore, N=158,010, and 158,010 divided by 1000 is 158 with a remainder of 10. So, the remainder is 10.But wait, I'm confused because the initial answer was 167,480, which is higher. Maybe I made a mistake in the number of partitions for each k.Wait, the number of ways to partition the board into k tiles is C(8, k-1). So, for k=3, it's C(8,2)=28, which is correct. For k=4, C(8,3)=56, correct. For k=5, C(8,4)=70, correct. For k=6, C(8,5)=56, correct. For k=7, C(8,6)=28, correct. For k=8, C(8,7)=8, correct. For k=9, C(8,8)=1, correct.So, the number of partitions is correct. Therefore, the issue must be with the colorings.Wait, perhaps I made a mistake in the inclusion-exclusion formula. The formula is 3^k - 3*2^k + 3*1^k. Wait, no, the inclusion-exclusion for exactly three colors is 3^k - 3*2^k + 3*1^k. But in our case, we want colorings where all three colors are used at least once, so it's 3^k - 3*2^k + 3*1^k. Wait, but 1^k is just 1, so it's 3^k - 3*2^k + 3.Yes, that's correct. So, for k=3, 3^3 - 3*2^3 +3=27-24+3=6, correct.Similarly, for k=4: 81 - 48 +3=36, correct.So, the colorings are correct.Therefore, my calculation of N=158,010 seems correct, leading to a remainder of 10 when divided by 1000.But wait, the initial answer was 167,480, which is different. So, perhaps I made a mistake in the initial step.Wait, let me re-examine the initial problem statement. It says that each tile may cover any number of consecutive squares, and each tile lies completely on the board. So, the number of ways to partition the board into k tiles is indeed C(8, k-1). That part is correct.Wait, but perhaps the initial answer had a different approach. Maybe they considered the number of tilings without color restrictions first, then applied inclusion-exclusion on the colorings.Let me try that approach.The total number of tilings without color restrictions is the sum over k=1 to 9 of C(8, k-1)*3^k.Then, the number of tilings using all three colors is total tilings minus tilings using only one or two colors.So, total tilings T = sum_{k=1}^9 C(8, k-1)*3^k.Number of tilings using only one color: For each color, the number of tilings is sum_{k=1}^9 C(8, k-1)*1^k = sum_{k=1}^9 C(8, k-1) = 2^8=256. Since there are 3 colors, it's 3*256=768.Number of tilings using exactly two colors: For each pair of colors, the number of tilings is sum_{k=1}^9 C(8, k-1)*2^k. The number of such pairs is C(3,2)=3.So, total tilings using exactly two colors is 3*(sum_{k=1}^9 C(8, k-1)*2^k).But wait, sum_{k=1}^9 C(8, k-1)*2^k = sum_{m=0}^8 C(8,m)*2^{m+1} }= 2*sum_{m=0}^8 C(8,m)*2^m }= 2*(1+2)^8=2*3^8=2*6561=13122.Wait, that seems high. Let me check:sum_{k=1}^9 C(8, k-1)*2^k = sum_{m=0}^8 C(8,m)*2^{m+1} }= 2*sum_{m=0}^8 C(8,m)*2^m }= 2*(1+2)^8=2*3^8=2*6561=13122.Yes, that's correct.Therefore, the number of tilings using exactly two colors is 3*13122=39366.But wait, this counts tilings where exactly two colors are used. However, we need to subtract tilings where only one color is used, which we've already counted as 768.Wait, no, in inclusion-exclusion, the formula is:Number of tilings using all three colors = Total tilings - tilings using at most two colors.But tilings using at most two colors include tilings using exactly one color and exactly two colors.So, the formula is:N = T - (tilings using one color + tilings using two colors).But tilings using two colors include tilings where both colors are used, but we have to be careful not to subtract too much.Wait, actually, the correct inclusion-exclusion formula is:Number of tilings using all three colors = Total tilings - 3*(tilings using one color) + 3*(tilings using two colors).Wait, no, that's not correct. The standard inclusion-exclusion for three sets is:|A ∪ B ∪ C| = |A| + |B| + |C| - |A∩B| - |A∩C| - |B∩C| + |A∩B∩C|.But in our case, we want the complement: tilings that use all three colors, which is total tilings minus tilings that use only one or two colors.So, it's:N = T - (tilings using only one color + tilings using only two colors).But tilings using only two colors can be calculated as C(3,2)*(tilings using exactly two colors). Wait, no, tilings using exactly two colors is C(3,2)*(tilings using two specific colors, excluding the third).But in our case, tilings using exactly two colors would be C(3,2)*(sum_{k=1}^9 C(8, k-1)*(2^k - 2)).Wait, no, that's not correct. The number of tilings using exactly two colors is C(3,2)*(sum_{k=1}^9 C(8, k-1)*(2^k - 2)). Because for each pair of colors, the number of tilings using both colors is 2^k - 2 (subtracting the cases where only one color is used).But this seems complicated. Alternatively, the number of tilings using exactly two colors is C(3,2)*(sum_{k=1}^9 C(8, k-1)*(2^k - 2)).But let's compute it step by step.First, total tilings T = sum_{k=1}^9 C(8, k-1)*3^k.Compute T:sum_{k=1}^9 C(8, k-1)*3^k = sum_{m=0}^8 C(8,m)*3^{m+1} }= 3*sum_{m=0}^8 C(8,m)*3^m }= 3*(1+3)^8=3*4^8.4^8=65536, so 3*65536=196,608.So, T=196,608.Now, number of tilings using only one color: 3*(sum_{k=1}^9 C(8, k-1)*1^k )=3*(sum_{m=0}^8 C(8,m)*1^{m+1})=3*(1+1)^8=3*256=768.Number of tilings using exactly two colors: For each pair of colors, the number of tilings is sum_{k=1}^9 C(8, k-1)*(2^k - 2). Because for each k, the number of tilings using both colors is 2^k - 2 (subtracting the two monochromatic tilings).So, for each pair, it's sum_{k=1}^9 C(8, k-1)*(2^k - 2)=sum_{k=1}^9 C(8, k-1)*2^k - 2*sum_{k=1}^9 C(8, k-1).We already know sum_{k=1}^9 C(8, k-1)*2^k=13122.And sum_{k=1}^9 C(8, k-1)=2^8=256.So, for each pair, it's 13122 - 2*256=13122 - 512=12610.Since there are C(3,2)=3 pairs, total tilings using exactly two colors is 3*12610=37,830.Therefore, the number of tilings using all three colors is:N = T - (tilings using one color + tilings using exactly two colors) = 196,608 - (768 + 37,830) = 196,608 - 38,598 = 158,010.Yes, that matches my earlier calculation. So, N=158,010.Therefore, the remainder when N is divided by 1000 is 158,010 mod 1000.To find this, divide 158,010 by 1000:158,010 ÷ 1000 = 158 with a remainder of 10.So, the remainder is 10.But wait, the initial answer was 167,480, which mod 1000 is 480. So, there must be a misunderstanding.Wait, perhaps the initial approach was incorrect because it didn't account for the fact that for each k, the number of colorings is 3^k - 3*2^k + 3, but when summing over all k, we have to ensure that we're not double-counting or missing something.But according to the inclusion-exclusion approach, N=158,010 is correct. Therefore, the remainder is 10.But let me check the initial approach again. The initial answer calculated N as 167,480, which is higher than 158,010. So, perhaps the initial approach had an error in the colorings for k=8 and k=9.Wait, in the initial answer, for k=8, it was 5610, but correct is 5796. Similarly, for k=9, it was 17,442, but correct is 18,150.So, the initial answer had:k=3:28*6=168k=4:56*36=2016k=5:70*150=10,500k=6:56*540=30,240k=7:28*1806=50,568k=8:8*5610=44,880k=9:1*17,442=17,442Summing these:168 + 2016 = 21842184 + 10,500 = 12,68412,684 + 30,240 = 42,92442,924 + 50,568 = 93,49293,492 + 44,880 = 138,372138,372 + 17,442 = 155,814Wait, that's 155,814, which is less than 167,480. Hmm, perhaps the initial answer had a different miscalculation.Wait, perhaps the initial answer incorrectly calculated the colorings for k=8 as 5610 instead of 5796, and for k=9 as 17,442 instead of 18,150. So, the difference is:For k=8: 5796 - 5610 = 186For k=9: 18,150 - 17,442 = 708So, the total difference is 186 + 708 = 894. Therefore, the initial answer was 155,814 + 894 = 156,708, which is still less than 167,480.Wait, perhaps the initial answer had a different approach altogether, such as considering the number of tilings as 3^9 - 3*2^9 + 3*1^9, which is 19,683 - 1,536 + 3 = 18,150, but that's only for k=9. So, that approach is incorrect because it doesn't account for all possible k.Therefore, the correct approach is the inclusion-exclusion over all k, leading to N=158,010, and the remainder modulo 1000 is 10.But wait, let me double-check the inclusion-exclusion formula again. The formula is:Number of colorings using all three colors = sum_{k=1}^9 C(8, k-1)*(3^k - 3*2^k + 3).Which is exactly what I did earlier, leading to N=158,010.Therefore, I think the initial answer had an error in the colorings for k=8 and k=9, leading to a higher N. The correct N is 158,010, and the remainder modulo 1000 is 10.But wait, let me check the initial answer again. It said:N = 28*6 + 56*36 + 70*150 + 56*540 + 28*1806 + 8*5610 + 1*17442 = 167,480.But according to my calculations, the correct colorings for k=8 and k=9 are 5796 and 18,150, respectively. So, the correct N should be:28*6=16856*36=201670*150=10,50056*540=30,24028*1806=50,5688*5796=46,3681*18,150=18,150Summing these:168 + 2016 = 21842184 + 10,500 = 12,68412,684 + 30,240 = 42,92442,924 + 50,568 = 93,49293,492 + 46,368 = 139,860139,860 + 18,150 = 158,010Yes, that's correct. So, the initial answer had incorrect colorings for k=8 and k=9, leading to a higher N. Therefore, the correct N is 158,010, and the remainder modulo 1000 is 10.But wait, let me confirm once more. 158,010 divided by 1000 is 158 with a remainder of 10. So, the remainder is 10.Therefore, the final answer is 10.

answer：First, I need to understand the dimensions of the rectangle and the circles. The rectangle measures 16 inches by 10 inches, and each circle has a diameter of 8 inches, which means each circle has a radius of 4 inches.To maximize the distance between the centers of the two circles, I should place them as far apart as possible within the rectangle. This typically involves positioning the circles in opposite corners of the rectangle.Next, I'll determine the coordinates of the centers of the circles. If I place one circle in the bottom-left corner, its center will be 4 inches from both the left and bottom edges of the rectangle. Similarly, placing the second circle in the top-right corner, its center will be 4 inches from both the right and top edges.Now, I'll calculate the horizontal and vertical distances between the centers of the two circles. The horizontal distance is the total width of the rectangle minus twice the radius of the circles, which is 16 inches minus 8 inches, resulting in 8 inches. The vertical distance is the total height of the rectangle minus twice the radius, which is 10 inches minus 8 inches, resulting in 2 inches.Finally, to find the greatest possible distance between the centers, I'll use the Pythagorean theorem to calculate the diagonal distance between the two centers. This involves taking the square root of the sum of the squares of the horizontal and vertical distances: √(8² + 2²) = √(64 + 4) = √68 = 2√17 inches.

answer：Okay, so I'm trying to figure out what x represents in this system of equations. The problem is about traveling from location A to location B, which has an uphill section and a flat section. The speeds given are 3 km/h uphill, 4 km/h on the flat, and 5 km/h downhill. It takes 36 minutes to go from A to B and 24 minutes to go back from B to A. The teacher provided these equations:[left{begin{array}{l}frac{x}{3} + frac{y}{4} = frac{36}{60} frac{x}{5} + frac{y}{4} = frac{24}{60}end{array}right.]Alright, so I need to understand what x and y stand for here. Let's break it down.First, the trip from A to B takes 36 minutes, which is 0.6 hours (since 36 divided by 60 is 0.6). The trip back from B to A takes 24 minutes, which is 0.4 hours. Looking at the first equation: (frac{x}{3} + frac{y}{4} = 0.6). This seems to represent the time taken to go from A to B. Since the uphill speed is 3 km/h, the time taken to cover the uphill distance x would be (frac{x}{3}). Similarly, the flat section has a speed of 4 km/h, so the time for the flat distance y is (frac{y}{4}). Adding these gives the total time from A to B.Now, the second equation: (frac{x}{5} + frac{y}{4} = 0.4). This is the time taken to go back from B to A. Here, what was uphill from A to B becomes downhill from B to A, and the speed for downhill is 5 km/h. So, the time for the downhill distance x is (frac{x}{5}). The flat section remains the same, so it's still (frac{y}{4}). Adding these gives the total time from B to A.So, putting it all together, x represents the distance of the uphill section from A to B, and y represents the distance of the flat section. That makes sense because when going back, the uphill becomes downhill, hence the different speed.I think I've got it. x is the uphill distance, and y is the flat distance. So, in the equations, x stands for the uphill section's length from A to B.

answer：First, I need to determine the range between (3^6 + 1) and (3^{12} + 1). Calculating these values:[3^6 = 729 quad Rightarrow quad 3^6 + 1 = 730][3^{12} = (3^6)^2 = 729^2 = 531441 quad Rightarrow quad 3^{12} + 1 = 531442]Next, I identify the smallest and largest perfect cubes within this range. The smallest perfect cube greater than 730 is (10^3 = 1000), and the largest perfect cube less than or equal to 531442 is (81^3 = 531441).To find the number of perfect cubes between these bounds, I count the integers from 10 to 81, inclusive:[81 - 10 + 1 = 72]Therefore, there are 72 perfect cubes between (3^6 + 1) and (3^{12} + 1), inclusive.

answer：Okay, so I need to find the domain of the function ( y = log_{0.5}(2x - 8) ). Hmm, I remember that for logarithmic functions, the argument inside the log has to be positive because you can't take the log of zero or a negative number. So, the expression inside the log, which is ( 2x - 8 ), must be greater than zero.Let me write that down: ( 2x - 8 > 0 ). Now, I need to solve this inequality for ( x ). To do that, I'll start by adding 8 to both sides to isolate the term with ( x ). So, adding 8 gives me ( 2x > 8 ).Next, I'll divide both sides by 2 to solve for ( x ). Dividing by 2, I get ( x > 4 ). So, this tells me that ( x ) has to be greater than 4 for the function to be defined.Wait, let me double-check. If ( x = 4 ), then ( 2x - 8 = 0 ), and the log of zero is undefined. So, ( x ) can't be equal to 4, only greater than 4. That makes sense.Is there anything else I need to consider? Well, the base of the logarithm is 0.5, which is between 0 and 1. I remember that the base of a logarithm can't be 1 or negative, but as long as it's positive and not equal to 1, it's okay. So, 0.5 is a valid base.Does the base affect the domain? I don't think so because the domain is determined by the argument of the logarithm, not the base itself. The base affects the behavior of the function, like whether it's increasing or decreasing, but not the domain.So, putting it all together, the domain of the function ( y = log_{0.5}(2x - 8) ) is all real numbers ( x ) such that ( x > 4 ). In interval notation, that would be ( (4, infty) ).Just to make sure, let me plug in a value greater than 4, say ( x = 5 ). Then, ( 2(5) - 8 = 10 - 8 = 2 ), and ( log_{0.5}(2) ) is defined because 2 is positive. What about a value less than 4, like ( x = 3 )? Then, ( 2(3) - 8 = 6 - 8 = -2 ), and ( log_{0.5}(-2) ) is undefined. So, my domain seems correct.I think I've covered all the necessary steps. The key was recognizing that the argument of the logarithm must be positive and solving the resulting inequality.