Bunuel wrote:

The length, width, and height of a rectangular box are in the ratio of 6:4:5, respectively. Express the height of the box in terms of its volume V.

A. \(\frac{1}{2}*\sqrt[3]{\frac{V}{3}}\)

B. \(\frac{5}{6}*\sqrt[3]{\frac{V}{15}}\)

C. \(2*\sqrt[3]{\frac{V}{15}}\)

D. \(\frac{5}{2}*\sqrt[3]{\frac{V}{15}}\)

E. \(3*\sqrt[3]{\frac{V}{15}}\)

As the calculation we're expected to do is straightforward, we'll just do it.

This is a Precise approach.

Writing V for volume and x for the height, we have:

\(V = x *( \frac{x}{5}*4) * (\frac{x}{5}*6) = \frac{24x^3}{25}\)

(Since the width is \(\frac{x}{5}*4\)and the length is \(\frac{x}{5}*6\))

Simplifying gives

\(x = \sqrt[3]{\frac{25V}{24}}=\sqrt[3]{\frac{5^{2}V}{3*2^{3}}}=\sqrt[3]{\frac{5^3V}{5*3*2^3}}=\frac{5}{2}*\sqrt[3]{\frac{V}{15}}\)

(D) is our answer.

_________________

David

Senior tutor at examPAL

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