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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
d. \(13 \sqrt{2}\)

Step by step solution

01

Compute the Width

Substitute the given height, length and volume into the formula of Volume = Length x Width x Height, and solve for the width. \n Given: Volume = 780, Length = 12, Height = 5 \n Substituting these values, we get \n 780 = 12 * Width * 5 \n Rearranging to find Width = 780 / (12 * 5) = 13.
02

Calculate the Diagonal

Now, we'll calculate the space diagonal using the width, height and length. For a rectangular solid, this diagonal distance ‘d’ can be found with the formula: \(d = \sqrt{{Width}^2 + {Height}^2 + {Length}^2}\) Substituting these values, we get \(d = \sqrt{{13}^2 + {5}^2 + {12}^2} = \sqrt{318} = 13√2\) which is the greatest distance between two vertices.

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