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question:The area of rectangle ABCD with vertices A(0, 0), B(0, 4), C(x, 4) and D(x, 0) is 28 square units. If x > 0, what is the value of x?
answer:The area of rectangle ABCD is given by the product of its length and width. The length of the rectangle is x - 0 = x units, and the width is 4 - 0 = 4 units. Therefore, the area of the rectangle is x cdot 4 = 4x square units. We are given that the area of the rectangle is 28 square units, so we can set up the equation 4x = 28. Dividing both sides of the equation by 4, we have x = frac{28}{4} = boxed{7}. The answer is: 7
question:A wooden model of a square pyramid has a base edge of 12 cm and an altitude of 8 cm. A cut is made parallel to the base of the pyramid that separates it into two pieces: a smaller pyramid and a frustum. Each base edge of the smaller pyramid is 6 cm and its altitude is X cm. There are 336 cubic centimeters in the volume of the frustum. What is the value of unknown variable X?
answer:To solve this problem, we need to determine the value of x, which represents the altitude of the smaller pyramid. The volume of a frustum can be calculated using the formula V = (1/3)h(A1 + A2 + √(A1 * A2)), where V is the volume, h is the altitude, A1 is the area of the larger base, and A2 is the area of the smaller base. We know the volume of the frustum is 336 cubic centimeters, and the altitude of the larger pyramid is 8 cm. The base edge of the larger pyramid is 12 cm, so the area of the larger base is (12^2) = 144 square centimeters. The base edge of the smaller pyramid is 6 cm, so the area of the smaller base is (6^2) = 36 square centimeters. Let's substitute these values into the volume formula: 336 = (1/3)x(144 + 36 + √(144 * 36)) 336 = (1/3)x(180 + √(5184)) 336 = (1/3)x(180 + 72) 336 = (1/3)x(252) To isolate x, we multiply both sides of the equation by 3/1: (3/1) * 336 = (3/1) * (1/3)x(252) 1008 = x(252) To solve for x, we divide both sides of the equation by 252: 1008 / 252 = x(252) / 252 4 = x The value of x is 4. The answer is: 4
question:What is the area enclosed by the graph of |x| + |2y| = X shown here? [asy] draw((0,-10)--(0,10),Arrows); draw((-15,0)--(15,0),Arrows); label("y",(0,10),NE); label("x",(15,0),SE); draw((10,0)--(0,5)--(-10,0)--(0,-5)--cycle); [/asy] If we know the answer to the above question is 100, what is the value of unknown variable X?
answer:We want to find the value of X in the given situation. The graph represents the area enclosed by the equation |x| + |2y| = X. To find the value of X, we can analyze the graph. From the graph, we can see that the shape enclosed by the equation is a square. The length of each side of the square is equal to X. Given that the area enclosed by the graph is 100, we can set up the equation: X^2 = 100 To find the value of X, we can take the square root of both sides of the equation: X = sqrt{100} Taking the square root of 100 gives us two possible values, +10 and -10. However, since we are looking for the length of a side of the square, X must be a positive value. Therefore, the value of X is 10. The answer is: 10
question:Sandra had 2 different bags of candy. Each of her bags had 6 pieces of candy left. Her brother, Roger, also had 2 bags of candy. One of his bags of candy had x pieces left and the other had 3 pieces left. How much more candy did Roger have? If we know the answer to the above question is 2, what is the value of unknown variable x?
answer:Sandra had 2 bags of candy, and each bag had 6 pieces of candy left, so she had a total of 2 * 6 = 12 pieces of candy. Roger also had 2 bags of candy. We know that one of his bags had x pieces left and the other had 3 pieces left. So he had a total of x + 3 pieces of candy. To find out how much more candy Roger had, we need to subtract the amount of candy Sandra had from the amount of candy Roger had: (x + 3) - 12. We are given that the difference is 2, so we can write: (x + 3) - 12 = 2. Simplifying the left side, we get: x + 3 - 12 = 2. Combining like terms, we get: x - 9 = 2. Adding 9 to both sides, we get: x = 11. The value of x is 11. 11 The answer is: 11