Chapter 2, Problem 44RE

To determine

To find: The instantaneous rate of change of the volume.

Answer to Problem 44RE

The instantaneous rate of change of the volume is $\frac{2}{3}\pi aH$ .

Explanation of Solution

Given information: The volume of cone is $V=\frac{1}{3}\pi r^{2}H$ , the point is $r=a$ and height does not change.

Formula used: The formula for average rate of change is given by,

  $\frac{\Delta y}{\Delta t}=\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}$

Calculation:

The rate of change of the volume can be calculated as:

  $\begin{array}{c}\frac{\Delta V}{\Delta r}=\underset{h\to 0}{\lim }\frac{\frac{1}{3}\pi {\left(r+h\right)}^{2}H-\frac{1}{3}\pi r^{2}H}{h}\\ =\underset{h\to 0}{\lim }\frac{\frac{1}{3}\pi H\left[{\left(r+h\right)}^2-r^2\right]}{h}\\ =\underset{h\to 0}{\lim }\frac{\frac{1}{3}\pi H\left[r^2+2rh+h^2-r^2\right]}{h}\\ =\underset{h\to 0}{\lim }\frac{\frac{1}{3}\pi H\left[2rh+h^2\right]}{h}\end{array}$

Solve further.

  $\frac{\Delta V}{\Delta r}=\underset{h\to 0}{\lim }\frac{\frac{1}{3}\pi hH\left[2r+h\right]}{h}$

Substitute $a$ for $r$ and 0 for $h$ in above equation.

  $\begin{array}{c}\frac{\Delta V}{\Delta r}=\underset{h\to 0}{\lim }\frac{1}{3}\pi H\left[2a+0\right]\\ =\frac{2}{3}\pi aH\end{array}$

Therefore, the instantaneous rate of change of volume is $\frac{2}{3}\pi aH$ .