Question

In Exercises 52-55, Linda is bored and decides to pour an entire container of salt into a pile on the kitchen floor. She pours 3 cubic inches of salt per second into a conical pile whose height is always two-thirds of its radius.

How fast is the radius of the conical salt pile changing when the radius of the pile is $2$inches?

Short answer

The rate of change is$\frac{27}{16\pi }\frac{\text{inches}}{\text{sec}}$

Step-by-step solution

Step 1. Given information

Given the radius is$2$inches and height is always two-thirds of its radius

Calculate the height and the rate of change

Calculating, we get

$$\begin{gathered} h=\frac{2}{3}r \\ h=\frac{2}{3}\times 2 \\ h=\frac{4}{3} \\ \frac{dV}{dt}=\frac{1}{3}\pi \times 2r\frac{dr}{dt}\times h \\ 3=\frac{1}{3}\pi \times 2\times 2\times \frac{dr}{dt}\times \frac{4}{3} \\ \frac{dr}{dt}=\frac{27}{16\pi }\frac{\text{inches}}{\sec } \end{gathered}$$