Question

The interior of a rectangular carton is designed by a certain manufacturer to have a volume of $x$ cubic feet and a ratio of length to width to height of $3:2:2$ . In terms of $x$, which of the following equals the height of the carton, in feet?

• A. $\sqrt[3]{3x}$
• B. $\sqrt[3]{3\frac{2x}{3}}$
• C. $\sqrt[3]{3\frac{3x}{2}}$
• D. $\frac{2}{3}\sqrt[3]{3x}$
• E. $\frac{3}{2}\sqrt[3]{3x}$

Answer 1

Answer: (B)

$\sqrt[3]{3\frac{2x}{3}}$

Let length, $l=3a$, breadth, $b=2a$, height $=2a$

$\therefore$ Volume $=lbh=x$ cubic feet

$\Rightarrow 3a\times 2a\times 2a=x\Rightarrow x=12a^3\Rightarrow a^3=\frac{x}{12}\Rightarrow a=\sqrt[3]{3\frac{x}{12}}$

Since, height $=2a$, in terms of $x$ the height $=2a=2$$\sqrt[3]{3\frac{x}{12}}$

$\Rightarrow \sqrt[3]{38}.\sqrt[3]{3\frac{x}{12}}=\sqrt[3]{3\frac{8x}{12}}=\sqrt[3]{3\frac{2x}{3}}$