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Difficulty: MediumVolume and Surface Area of Solids

A solid rectangular prism has a volume of 192192 cubic inches. The ratio of the length of the prism to its width is 3:13:1, and the height of the prism is 44 inches. What is the surface area, in square inches, of the prism?

Answer: 224 square inches

Answer

The surface area of the prism is 224 square inches.
The correct answer is 224. By using the volume formula V=lwhV = lwh with a volume of 192192 and a height of 44, the product of the length and width of the base is lw=48lw = 48. Given that the ratio of the length to the width is 3:13:1, the length can be written as l=3wl = 3w. Substituting this expression into the product equation gives 3w(w)=483w(w) = 48, or 3w2=483w^2 = 48. Dividing both sides by 3 yields w2=16w^2 = 16, which means the width of the prism is 44 inches. Since l=3wl = 3w, the length is 1212 inches. The surface area of the prism can be found using the formula SA=2(lw+lh+wh)SA = 2(lw + lh + wh). Substituting the dimensions l=12l = 12, w=4w = 4, and h=4h = 4 yields SA=2(12⋅4+12⋅4+4⋅4)=2(48+48+16)=2(112)=224SA = 2(12 \cdot 4 + 12 \cdot 4 + 4 \cdot 4) = 2(48 + 48 + 16) = 2(112) = 224 square inches.

Step-by-Step Solution

1
Use the volume formula for a rectangular prism, V=lwhV = lwh, and substitute the given volume of 192192 and height of 44.
192=l⋅w⋅4192 = l \cdot w \cdot 4, which simplifies to l⋅w=48l \cdot w = 48.
To find the product of the length and width of the base of the prism.
2
Express the length in terms of the width using the ratio of 3:13:1.
l=3wl = 3w.
The ratio of the length to the width is given as 3 to 1.
3
Substitute l=3wl = 3w into the equation l⋅w=48l \cdot w = 48 and solve for ww.
3w⋅w=48  ⟹  3w2=48  ⟹  w2=16  ⟹  w=43w \cdot w = 48 \implies 3w^2 = 48 \implies w^2 = 16 \implies w = 4.
To solve for the width of the rectangular prism.
4
Find the length of the rectangular prism.
l=3(4)=12l = 3(4) = 12 inches.
Since the length is 3 times the width and the width is 4 inches, the length must be 12 inches.
5
Use the surface area formula SA=2(lw+lh+wh)SA = 2(lw + lh + wh) with the dimensions l=12l = 12, w=4w = 4, and h=4h = 4.
SA=2(12⋅4+12⋅4+4⋅4)=2(48+48+16)=2(112)=224SA = 2(12 \cdot 4 + 12 \cdot 4 + 4 \cdot 4) = 2(48 + 48 + 16) = 2(112) = 224.
To calculate the total surface area of the prism.

Key Concept

Volume and Surface Area of Rectangular Prisms
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Topics

Volume and Surface Area of Solids

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