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Chapter 14: Problem 18

What is the greatest distance between two vertices of a rectangular solid with a height of \(5,\) a length of \(12,\) and a volume of \(780 ?\) a. 12 b. \(12 \sqrt{2}\) c. 13 d. \(13 \sqrt{2}\) e. \(13 \sqrt{3}\)

Short Answer

Expert verified
b. \( 13 \sqrt{2} \) units

Step by step solution

01

Calculate the Width

Given the volume, height, and length, calculate the width (W) of the rectangular solid using the formula for volume. Solving the equation \( 5 * 12 * W = 780 \) for W, we get \( W = \frac{780}{60} = 13 \) units.
02

Calculate the Space Diagonal

From the Pythagorean theorem in three dimensions, the space diagonal can be calculated using the lengths of all the sides. The space diagonal (D) is given by the formula \( D = \sqrt{H^2 + L^2 + W^2} \), where H is the height, L is the length, and W is the width. Substituting the given values, \( D = \sqrt{5^2 + 12^2 + 13^2} = \sqrt{25 + 144 + 169} = \sqrt{338} \).
03

Simplify Radical Form

Simplify the radical form \(\sqrt{338} \) to find the answer. Since \( 338 = 2 * 13^2 \), we can rewrite the radical as \( \sqrt{2 * 169} = 13 \sqrt{2} \) units.

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