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question:f Bailey is using a 100cm length of rope to create a rectangle, and the longer sides of the rectangle measure 28cm each, what is the length of each of the shorter sides?
answer:The perimeter of a rectangle is given by the formula P = 2L + 2W, where P is the perimeter, L is the length, and W is the width. In this case, we know that the perimeter is 100cm and the longer sides (length) measure 28cm each. So, we can set up the equation 100 = 2(28) + 2W. Simplifying, we get 100 = 56 + 2W. Subtracting 56 from both sides, we get 44 = 2W. Dividing both sides by 2, we get W = 22. Therefore, each of the shorter sides of the rectangle measures 22cm. 22 The answer is: 22
question:Suppose the function f has all real numbers in its domain and range and is invertible. Some values of f are given by the following table: begin{array}{c || c | c | c | c | c} x & 1 & 2 & 3 & 4 & 5 hline f(x) & X & 3 & 5 & 7 & 8 end{array}What is the value of f(f(3)) + f(f^{-1}(4)) + f^{-1}(f^{-1}(5))? If there is not enough information to answer this question, enter "NEI". If we know the answer to the above question is 14, what is the value of unknown variable X?
answer:We are given a table that shows the values of f for certain values of x. We are asked to find the value of f(f(3)) + f(f^{-1}(4)) + f^{-1}(f^{-1}(5)). To find the value of f(f(3)), we first need to find the value of f(3). From the table, we see that f(3) = 5. So, f(f(3)) = f(5). To find the value of f(f^{-1}(4)), we need to find the value of f^{-1}(4). The inverse of f is denoted as f^{-1}. From the table, we see that f(4) = 7. So, f^{-1}(4) = 7. To find the value of f^{-1}(f^{-1}(5)), we need to find the value of f^{-1}(5). From the table, we see that f(5) = 8. So, f^{-1}(5) = 8. Now, we can substitute these values into the expression: f(f(3)) + f(f^{-1}(4)) + f^{-1}(f^{-1}(5)) = f(5) + f(7) + f^{-1}(8). From the table, we see that f(5) = 8 and f(7) = 4. So, the expression becomes: 8 + 4 + f^{-1}(8). We are given that the value of the expression is 14. So, we can write: 8 + 4 + f^{-1}(8) = 14. Subtracting 12 from both sides of the equation, we find: f^{-1}(8) = 2. The inverse of f is denoted as f^{-1}. So, f^{-1}(8) = X. Given that the value of X is 2, we can substitute it into the equation: 2 = 2. The value of X is 2. The answer is: 2
question:Lily bought a Russian nesting doll as a souvenir. The biggest doll is 243 cm tall, and each doll is 2/3rd the size of the doll that contains it. How big is the 6th biggest doll?
answer:The biggest doll is 243 cm tall. The second biggest doll is 2/3 * 243 cm = 162 cm tall. The third biggest doll is 2/3 * 162 cm = 108 cm tall. The fourth biggest doll is 2/3 * 108 cm = 72 cm tall. The fifth biggest doll is 2/3 * 72 cm = 48 cm tall. The sixth biggest doll is 2/3 * 48 cm = 32 cm tall. Therefore, the 6th biggest doll is 32 cm tall. 32 The answer is: 32
question:Jess and her family play Jenga, a game made up of 54 stacked blocks in which each player removes one block in turns until the stack falls. The x players, including Jess, play 5 rounds in which each player removes one block. In the sixth round, Jess's father goes first. He removes a block, causing the tower to almost fall. Next, Jess tries to remove another block knocking down the tower. How many blocks did the tower have before Jess's turn? If we know the answer to the above question is 28, what is the value of unknown variable x?
answer:We know that in each round, including the sixth round, each player removes one block. In the first five rounds, there are x players, including Jess, so the total number of blocks removed is 5 * x. In the sixth round, before Jess's turn, her father removes a block, so the total number of blocks removed is 5 * x + 1. We are given that the total number of blocks before Jess's turn is 28, so we can write: 54 - (5 * x + 1) = 28. Simplifying the left side, we get: 54 - 5x - 1 = 28. Combining like terms, we get: 53 - 5x = 28. Subtracting 53 from both sides, we get: -5x = -25. Dividing both sides by -5, we get: x = 5. The value of x is 5. 5 The answer is: 5