Question

in a rectangular parallelepiped the length and width of the base are 12 in. and 9 in respectively. find the volume of the solid if the length of the diagonal of the solid is 25 inches.

Answer 1

Answer on Question #64761 – Math – Geometry

Question

In a rectangular parallelepiped the length and width of the base are 12 in. and 9 in respectively. find the volume of the solid if the length of the diagonal of the solid is 25 inches.

Solution

Given

$$AB = CD = A_1B_1 = C_1D_1 = 9 \text{ in.}, \quad AD = BC = A_1D_1 = B_1C_1 = 12 \text{ in.}, \quad B_1D = 25 \text{ in.}$$

We can find the bases diagonal from the right triangle ABD using the Pythagorean theorem:

$$BD = \sqrt{AB^2 + AD^2} = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \text{ in.}$$

Angle $B_1BD$ is right, applying the Pythagorean theorem to the right triangle $BB_1D$ compute

$$BB_1 = \sqrt{B_1D^2 - BD^2} = \sqrt{25^2 - 15^2} = \sqrt{625 - 225} = \sqrt{400} = 20 \text{ in.}$$

Thus, the volume of the solid is

$$V = AB \cdot BC \cdot BB_1 = 9 \cdot 12 \cdot 20 = 2160 \text{ in}^3.$$

Answer: $V = 2160 \text{ in}^3$.

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