answer：Okay, so I have this problem about a rhombus with diagonals of 10 inches and 24 inches, and I need to find the perimeter. Hmm, I remember that a rhombus is a type of quadrilateral where all sides are equal in length. That means if I can find the length of one side, I can multiply it by four to get the perimeter. First, I think about the properties of a rhombus. I recall that the diagonals of a rhombus bisect each other at right angles. So, if the diagonals are 10 inches and 24 inches, they cut each other exactly in half. That means each half of the diagonals would be 5 inches and 12 inches, respectively. Now, since the diagonals intersect at right angles, they form four congruent right-angled triangles inside the rhombus. Each of these triangles has legs measuring 5 inches and 12 inches. The hypotenuse of each triangle would be the side of the rhombus. To find the length of the side, I can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, if I let 's' be the length of the side of the rhombus, then:s² = 5² + 12²Calculating that, 5 squared is 25, and 12 squared is 144. Adding those together gives:s² = 25 + 144 = 169Taking the square root of both sides gives:s = √169 = 13 inchesSo, each side of the rhombus is 13 inches long. Since all sides are equal, the perimeter is just four times the length of one side:Perimeter = 4 × 13 = 52 inchesI think that makes sense. Let me double-check my steps. I used the fact that the diagonals bisect each other at right angles, which is a property of rhombuses. Then, I applied the Pythagorean theorem correctly to find the side length. Multiplying by four gives the total perimeter. Yeah, that seems right.Just to visualize, if I imagine the rhombus split into four right triangles by its diagonals, each triangle has legs of 5 and 12 inches. The hypotenuse, which is the side of the rhombus, is 13 inches. So, four sides of 13 inches each would indeed give a perimeter of 52 inches. I don't think I made any mistakes here. The calculations are straightforward, and the properties I used are correct for a rhombus. So, I'm confident that the perimeter is 52 inches.

answer：Alright, so I have this problem where Ben throws five identical darts at five identical dartboards, and I need to figure out how many different sequences can be formed by listing the number of darts on each board from greatest to least. Hmm, okay. Let me try to break this down step by step.First, since both the darts and the dartboards are identical, the order in which the darts land on the boards doesn't matter. What matters is how many darts end up on each board. So, essentially, I'm looking for the number of ways to distribute five identical items (darts) into five identical containers (dartboards). This sounds like a problem related to partitions in mathematics. A partition of a number is a way of writing it as a sum of positive integers, where the order of the addends doesn't matter. In this case, I want to partition the number 5 (the total number of darts) into up to five parts (since there are five dartboards). Each part represents the number of darts on a dartboard.Let me list out all the possible partitions of 5:1. **5**: This means all five darts land on a single dartboard, and the other four dartboards have zero darts. So, the sequence would be (5, 0, 0, 0, 0).2. **4 + 1**: Here, four darts land on one dartboard, and one dart lands on another. The remaining three dartboards have zero darts. The sequence would be (4, 1, 0, 0, 0).3. **3 + 2**: Three darts on one dartboard and two darts on another. The remaining two dartboards have zero darts. The sequence is (3, 2, 0, 0, 0).4. **3 + 1 + 1**: Three darts on one dartboard, and one dart each on two other dartboards. The remaining two dartboards have zero darts. The sequence would be (3, 1, 1, 0, 0).5. **2 + 2 + 1**: Two darts on two different dartboards and one dart on another. The remaining two dartboards have zero darts. The sequence is (2, 2, 1, 0, 0).6. **2 + 1 + 1 + 1**: Two darts on one dartboard and one dart each on three other dartboards. The remaining dartboard has zero darts. The sequence would be (2, 1, 1, 1, 0).7. **1 + 1 + 1 + 1 + 1**: One dart on each of the five dartboards. The sequence is (1, 1, 1, 1, 1).Let me double-check to make sure I haven't missed any partitions. Starting from the largest possible number of darts on a single dartboard and working my way down:- 5: Covered.- 4 + 1: Covered.- 3 + 2: Covered.- 3 + 1 + 1: Covered.- 2 + 2 + 1: Covered.- 2 + 1 + 1 + 1: Covered.- 1 + 1 + 1 + 1 + 1: Covered.I don't think there are any other partitions of 5 into up to five parts. Each partition corresponds to a unique sequence when ordered from greatest to least. Therefore, the number of different lists possible is equal to the number of partitions I've listed.So, counting them up: 1, 2, 3, 4, 5, 6, 7. That's seven different partitions, which means there are seven different sequences possible.Just to be thorough, let me consider if there's any overlap or if I've counted any partition more than once. Looking back:- (5, 0, 0, 0, 0) is unique.- (4, 1, 0, 0, 0) is unique.- (3, 2, 0, 0, 0) is unique.- (3, 1, 1, 0, 0) is unique.- (2, 2, 1, 0, 0) is unique.- (2, 1, 1, 1, 0) is unique.- (1, 1, 1, 1, 1) is unique.No overlaps or duplicates here. Each sequence is distinct and represents a different way the darts can be distributed across the dartboards.Therefore, I'm confident that the number of different lists possible is seven.

answer：First, I recognize that the area of a circle is given by the formula ( text{Area} = pi r^2 ), where ( r ) is the radius.Given the area is 132.73228961416876 square meters, I can rearrange the formula to solve for the radius:[ r^2 = frac{text{Area}}{pi} ]Substituting the given area:[ r^2 = frac{132.73228961416876}{pi} ]Calculating the value:[ r^2 approx frac{132.73228961416876}{3.14159} approx 42.2372956 ]Taking the square root to find the radius:[ r approx sqrt{42.2372956} approx 6.499022 ]Finally, to find the diameter, I double the radius:[ text{Diameter} = 2r approx 2 times 6.499022 approx 12.998044 ]

answer：First, I need to determine the total time Mark spends on activities other than kneading. He lets the bread rise twice, each time for 120 minutes, so that's 120 minutes multiplied by 2, which equals 240 minutes. Additionally, he spends 30 minutes baking the bread. Adding these together, the total time spent on rising and baking is 240 minutes plus 30 minutes, totaling 270 minutes. Given that the entire process takes 280 minutes, I can find the time spent kneading by subtracting the time spent on rising and baking from the total time. So, 280 minutes minus 270 minutes equals 10 minutes. Therefore, Mark spends 10 minutes kneading the bread.

answer：To convert the decimal number 72 into an octal number, I will use the division by 8 method.First, I divide 72 by 8, which gives a quotient of 9 and a remainder of 0.Next, I divide the quotient 9 by 8, resulting in a quotient of 1 and a remainder of 1.Finally, I divide the quotient 1 by 8, obtaining a quotient of 0 and a remainder of 1.By collecting the remainders from the last division to the first, the octal representation of 72 is 110.Therefore, the correct option is C.

answer：Alright, so I have this problem about three trains starting at the same time. Two of them are starting from towns that are 450 miles apart on parallel tracks, and one of them is going 6 mph faster than the other. Then there's a third train that starts from a town exactly in the middle of these two towns, but on a track that's perpendicular to the other two tracks. This third train meets the other two trains at the same spot after 5 hours. I need to find the speed of each of the three trains.Okay, let's break this down. First, the two trains on parallel tracks are 450 miles apart and are moving towards each other, right? One is faster than the other by 6 mph. So, if I let the speed of the slower train be 'v' mph, then the faster one would be 'v + 6' mph. Since they're moving towards each other, their speeds add up when calculating the time it takes for them to meet.They meet after 5 hours, so the total distance they cover together in that time should be 450 miles. That gives me the equation:5v + 5(v + 6) = 450Simplifying that:5v + 5v + 30 = 45010v + 30 = 45010v = 420v = 42So, the slower train is going at 42 mph, and the faster one is going at 48 mph.Now, the third train starts from the midpoint between the two towns, which is 225 miles from each town. It meets the other two trains after 5 hours as well. So, the third train must cover 225 miles in 5 hours. To find its speed, I can use the formula:Speed = Distance / TimeSo,Speed = 225 / 5 = 45 mphTherefore, the third train is going at 45 mph.Wait a minute, does this make sense? The third train is going slower than the faster train but faster than the slower one. That seems reasonable because it's starting from the midpoint and needs to reach the meeting point in the same time as the other two trains.Let me double-check the calculations. For the two parallel trains:5 hours * 42 mph = 210 miles5 hours * 48 mph = 240 miles210 + 240 = 450 milesThat checks out. For the third train:5 hours * 45 mph = 225 milesWhich is exactly half of 450 miles, so it makes sense that it starts from the midpoint and meets them at the meeting point.I think that's all. The speeds are 42 mph, 48 mph, and 45 mph for the three trains respectively.