The width of a rectangular solid is twice the height of the solid, and the height of the solid is twice the length of the solid. If x is the length of the solid, what is the surface area of the solid in terms of x?
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Solution
The surface area of a solid is the sum of the areas of each side of the solid. A rectangular solid has 6 rectangular faces. If x is the length of the solid, then 2x is the height of the solid and 4x is the width of the solid. Two faces of the solid measure x units by 2x units, two faces measure x units by 4x units, and two faces measures 2x units by 4x units. Therefore, the surface area of the solid is equal to 2(x 2x) + 2(x 4x) + 2(2x 4x) = 2(2x2) + 2(4x2) + 2(8x2) = 4x2 + 8x2 + 16x2 = 28x2
The volume of a cube is x3 cubic units, and the surface area of the cube is x3 square units. What is the value of x?
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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 the cube is equal to the cube root of x3, which is equal to x, since (x)(x)(x) = x3. 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 x, the area of any one face of the cube is (x)(x) = x2. A cube has six faces, so the total surface area of the cube is equal to 6x2 square units. It is given that the surface area of the square is x3 square units. Therefore, 6x2 = x3. Divide both sides by x2, and the value of x is 6.
The surface area of a rectangular solid is 192 cm2.If the height of the solid is 4 units and the length of the solid is 12 units, what is the width of the solid?
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Solution
The surface area of a solid is the sum of the areas of each side of the solid. A rectangular solid has 6 rectangular faces. If w is the width of the solid, then two faces measure 4 units by 12 units, two faces measure 4 units by w units, and two faces measures 12 units by w units. Therefore, the surface area of the solid is equal to 2(4 12) + 2(4 w) + 2(12 w) = 96 + 8w + 24w = 96 + 32w. Since the surface area of the solid is 192 cm2, 96 + 32w = 192, 32w = 96, w = 3. The width of the solid is 3 units.
Danielle’s cube has a volume of 512 in.3. What is the surface area 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 Danielle’s cube is equal to the cube root of 512, which is equal to 8, since (8)(8)(8) = 512. 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 8 units, the area of one face of the cube is (8)(8) = 64 square units. A cube has six faces, so the total surface area of the cube is equal to (64)(6) = 384 square units.
A rectangular solid measures 4 units by 5 units by 6 units. What is the surface area of the solid?
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Solution
The surface area of a solid is the sum of the areas of each side of the solid. A rectangular solid has 6 rectangular faces. Two faces measure 4 units by 5 units, two faces measure 4 units by 6 units, and two faces measure 5 units by 6 units. Therefore, the surface area of the solid is equal to 2(4 5) + 2(4 6) +2(5 6) = 2(20) + 2(24) + 2(30) = 40 + 48 + 60 = 148 square units.