Question

Chloe is a pharmacist and wants to know the
height of a medicine bottle received from a
supplier. The supplier informed her that the
radius of the medicine bottle is one-fourth its
height. Chloe knows that the medicine bottle
has a volume of V cubic inches.
Which of the following functions would best model the situation above?
O
cube root
O
square root
O
exponential
O
step

Answer 1

The given situation would be a best model of cube root.

Solution:

Generally, medicine bottles will be cylindrical in shape. Since the shape of the bottle is not given we can assume that the bottle is in cylindrical shape.

Given information:

Radius of the bottle (r) = one fourth of its height (h)

$\Rightarrow r=\frac{1}{4}\times h$

Volume of the bottle = V cubic inches

The formula of the volume of a cylinder is as follows,

$\Rightarrow V=\pi r^{2} h$

This can be re-written as $h=\frac{V}{\pi r^{2}}$ as we do not know the height.

On plugging-in the given values we get,

$\Rightarrow h=\frac{V}{\pi \times(\frac{h}{4})^{2}}\rightarrow \frac{V}{\pi \times\frac{h^2}{16}}\rightarrow \frac{16V}{\pi\times h^2}$

On solving we get,

$\Rightarrow h\times h^2 = \frac{V}{\pi}$

$\Rightarrow h^3 = \frac{V}{\pi}$

On taking the cube root on both sides we get,

$\Rightarrow h=\sqrt[3]{\frac{V}{\pi}}$

Therefore, cube root would be the best model of the given solution.