Q. 52 - Calculus

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?

Step-by-Step Solution

Answer

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

Step 1. Given information

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

Step 2: Calculate the height and the rate of change

Calculating, we get

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