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Zorluk: OrtaThree-Dimensional Geometry: Volume and Surface Area

Three adjacent faces of a right rectangular prism have surface areas of 2424, 3636, and 5454 square centimeters. What is the volume of the prism, in cubic centimeters?

Cevap: 216 cubic centimeters

Cevap

The volume of the rectangular prism is 216216 cubic centimeters.
Let the edge lengths of the right rectangular prism be ll, ww, and hh. The areas of three adjacent faces meeting at a common vertex are given by lw=24lw = 24, wh=36wh = 36, and lh=54lh = 54. Multiplying these three equations together yields (lw)(wh)(lh)=(lwh)2=24×36×54=46,656(lw)(wh)(lh) = (lwh)^2 = 24 \times 36 \times 54 = 46,656. Since the volume VV of a right rectangular prism is equal to lwhlwh, taking the positive square root gives V=46,656=216V = \sqrt{46,656} = 216 cubic centimeters.

Adım Adım Çözüm

1
Define variables for the three dimensions of the rectangular prism.
Let ll, ww, and hh represent the length, width, and height of the prism.
The area of each rectangular face is the product of two adjacent dimensions.
2
Express the given face areas in terms of the dimensions.
lw=24lw = 24, wh=36wh = 36, and lh=54lh = 54.
Adjacent faces share pairwise dimensions.
3
Multiply all three face area equations together.
(lw)(wh)(lh)=l2w2h2=(lwh)2=24×36×54=46,656(lw)(wh)(lh) = l^2 w^2 h^2 = (lwh)^2 = 24 \times 36 \times 54 = 46,656.
Multiplying the three face areas yields the square of the prism's volume (V=lwhV = lwh).
4
Calculate the square root of the product to determine the volume.
V=lwh=46,656=216V = lwh = \sqrt{46,656} = 216.
Taking the square root gives the volume directly without needing to solve for individual edge lengths.

Anahtar Kavram

For a right rectangular prism with adjacent face areas A1A_1, A2A_2, and A3A_3, the volume is given by V=A1×A2×A3V = \sqrt{A_1 \times A_2 \times A_3}.
Tahmini Süre:1m 15s
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Three-Dimensional Geometry: Volume and Surface Area

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