Chapter 2.6, Problem 24AYU

To determine

To express: the volume $V$ of the water in the cone as a function of the height $h$ of the water.

Answer to Problem 24AYU

The volume is $\frac{\pi h^2}{48}$ .

Explanation of Solution

Given information:

A container in the shape of a right circular cone with radius 4 feet and height 16 feet.

  

Formula Used:

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

Calculation:

Consider the following figure.

From the similar triangles ABC and APQ,

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

The volume $V$ is

  $\begin{array}{c}V=\frac{1}{3}\pi r^{2}h\\ V\left(h\right)=\frac{1}{3}\pi {\left(\frac{h}{4}\right)}^{2}h\\ =\frac{\pi h^2}{48}\end{array}$

So, the volume is $\frac{\pi h^2}{48}$ .