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 is 3:2:2. In terms of x, which of the following equals the height of the carton, in feet?
A. $ \sqrt[3]{x} $
B. $ \sqrt[3]{{\dfrac{{2x}}{3}}} $
C. $ \sqrt[3]{{\dfrac{{3x}}{2}}} $
D. $ \dfrac{2}{3}\sqrt[3]{x} $
E. $ \dfrac{3}{2}\sqrt[3]{x} $

Answer 1

Hint: We are given that the ratio of length to width to height of a rectangular carton is 3:2:2. Consider the length, width and height as 3a feet, 2a feet and 2a feet respectively. Find its volume using volume of a cuboid formula. Equate this obtained volume with x, because given that the volume of the rectangular carton is equal to x cubic feet. From this find the value of a in terms of x. The height is considered to be 2a feet, so substitute the obtained value of a in terms of x to find the height of the carton.
Formula used:
Volume of a cuboid with length l, height h and width w is $ l \times h \times w $

Complete step by step solution:
We are given that the volume of a rectangular carton is x cubic feet and the ratio of its length to width to height is 3:2:2. We have to find its height in terms of x.
Let the length be 3a feet, width be 2a feet and height be 2a feet.

Volume of the carton is $ l \times h \times w $
 $ \to 3a \times 2a \times 2a = 12{a^3} $ cubic feet.
We are given that this volume is also equal to x cubic feet.
This means,
 $ \Rightarrow 12{a^3} = x $
 $ \Rightarrow {a^3} = \dfrac{x}{{12}} $
 $ \Rightarrow a = \sqrt[3]{{\dfrac{x}{{12}}}} $
We have got the value of a.
Height of the carton is 2a feet, this means height is $ 2 \times \sqrt[3]{{\dfrac{x}{{12}}}} = \sqrt[3]{{\dfrac{{8x}}{{12}}}} = \sqrt[3]{{\dfrac{{2x}}{3}}} $ feet.
Hence the correct option is Option B, $ \sqrt[3]{{\dfrac{{2x}}{3}}} $ .
So, the correct answer is “Option B”.

Note: We have considered the carton as a cuboid because the given ratios of length, height and width, 3:2:2, are not equal. If the ratio is like 2:2:2, then we have to consider it as a cube, because length, height and width are equal in a cube and not equal in a cuboid.