The length of a rectangular solid is 6 units, and the height of the solid is 12 units. If the volume of the solid is 36 cubic units, what is the width of the solid?
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Solution
The volume of a rectangular solid is equal to lwh, where l is the length of the solid, w is the width of the solid, and h is the height of the solid. Therefore, (6)(12)(w) = 36, 72w = 36, and w = 1⁄2 units.
The volume of Stephanie’s cube is equal to 64x6.What is the area of one face of her cube?
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Solution
The volume of a cube is equal to the product of its length, width, and height. Since the length, width, and height of a cube are identical in measure, the measure of one edge of Stephanie’s cube is equal to the cube root of 64x6, which is equal to 4x2, since (4x2)(4x2)(4x2) = 64x6. The area of one face of the cube is equal to the product of the length and width of that face. Since every length and width of the cube is 4x2, the area of any one face of the cube is (4x2)(4x2) = 16x4.
The volume of rectangular solid A is equal to the volume of rectangular solid B. If the length of solid A is three times the length of solid B, and the height of solid A is twice the height of solid B, then
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Solution
The volume of a rectangular solid is equal to lwh, where l is the length of the solid, w is the width of the solid, and h is the height of the solid. If l is the length of solid B and h is the height of solid B, then the length of solid A is 3l and the height of solid A is 2h. Since the volumes of the solids are equal, if w1 represents the width of solid A and w2 represents the width of solid B, then (3l)(2h)(w1) = (l)(h)(w2), 6w1 = w2, which means that w2, the width of solid B, is equal to 6 times the width of solid A.
The area of one face of a cube is 9x square units.What is the volume of the cube?
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Solution
One face of a cube is a square. The area of a square is equal to the length of one side of the square multiplied by itself. Therefore, the length of a side of this square (and edge of the cube) is equal to \(\sqrt{9x}\), or 3√x units. Since every edge of a cube is equal in length and the volume of a cube is equal to e3, where e is the length of an edge (or lwh, l is the length of the cube, w is the width of the cube, and h is the height of the cube, which in this case, are all 3√x units), the volume of the cube is equal to (3√x)3 = 27x√x cubic units.
The length of a rectangular solid is twice the sum of the width and height of the rectangular solid. If the width is equal to the height and the volume of the solid is 108 in.3, what is the length of the solid?
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Solution
The volume of a rectangular solid is equal to lwh, where l is the length of the solid, w is the width of the solid, and h is the height of the solid. If x represents the width (and therefore, the height as well), then the length of the solid is equal to 2(x + x), or 2(2x) = 4x. Therefore, (4x)(x)(x) = 108, 4x3 = 108, x3 = 27, and x = 3. If the width and height of the solid are each 3 in, then the length of the solid is 2(3 + 3) = 2(6) = 12 in.
The height of a cylinder is four times the radius of the cylinder. If the volume of the cylinder is 256π cm3, what is the radius of the cylinder?
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Solution
The volume of a cylinder is equal to πr2h, where r is the radius of the cylinder and h is the height of the cylinder. Since the height is 4 times the radius, the volume of this cylinder is equal to πr2(4r) = 256π, 4r3 = 256, r3 = 64, r = 4. The radius of the cylinder is 4 cm.
The radius of a cylinder is 2x and the height of the cylinder is 8x + 2. What is the volume of the cylinder in terms of x?
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Solution
The volume of a cylinder is equal to πr2h, where r is the radius of the cylinder and h is the height of the cylinder. The volume of this cylinder is equal to π(2x)2(8x + 2) = π4x2(8x + 2) = (32x3 + 8x2)π.
The height of cylinder B is three times the height of cylinder A, and the radius of cylinder B is 1⁄3 the radius of cylinder A. Which of the following statements is true?
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Solution
The volume of a cylinder is equal to πr2h, where r is the radius of the cylinder and h is the height of the cylinder. If the volume of cylinder A is
πr2h, then the volume of cylinder B is π(1⁄3(r))2(3h) = π((1⁄9)r2)(3h) = π(1⁄3r)2h, which is 1⁄3 the volume of cylinder A.
Terri fills with water 2⁄3 of a glass that is 15 cm tall. If the radius of the glass is 2 cm, what volume of water is in Terri’s glass?
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Solution
The volume of a cylinder is equal to πr2h, where r is the radius of the cylinder and h is the height of the cylinder. Since Terri’s glass is only 2⁄3 full,the height of the water is 2⁄3(15) = 10 cm. Therefore, the volume of water is equal to: π(2)2(10) = 40π cm3.
A cylinder has a volume of 45π in.3. Which of the following could be the radius and height of the cylinder?
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Solution
The volume of a cylinder is equal to πr2h, where r is the radius of the cylinder and h is the height of the cylinder. Only the values of the radius and height given in choice a hold true in the formula: π(3)2(5) = π(9)(5) = 45π in.3.