Q. 24 - Precalculus Enhanced with Graphing Utilities

Question

Water is poured into a container in the shape of a right circular cone with radius $4$ feet and height $16$ feet. See the figure. Express the volume V of the water in the cone as a function of the height h of the water.

[Hint: The volume V of a cone of radius r and height h is $V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$.]

Step-by-Step Solution

Answer

The volume V of the water in the cone as a function of the height h of the water is $V=\frac{{\mathrm{\pi h}}^3}{48}$.

Step 1. Use the property of similar triangles.

We have given two similar triangles which gives

$$\begin{gathered} \frac{r}{h}=\frac{4}{16} \\ \frac{r}{h}=\frac{1}{4} \\ r=\frac{h}{4} \end{gathered}$$

Step 2. Use the formula for the volume of a cone.

The volume V of a cone is $V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$.

So the volume of cone as a function of height h is,

$$\begin{gathered} V=\frac{1}{3}\mathrm{\pi }{\left(\frac{\mathrm{h}}{4}\right)}^2\mathrm{h}=\frac{{\mathrm{\pi h}}^3}{3\left(16\right)} \\ =\frac{{\mathrm{\pi h}}^3}{48} \end{gathered}$$