Question

A heap of rubbish in the shape of a cube is being compacted into a smaller cube. Given that the volume decreases at a rate of 3 cubic meters per minute, find the rate of change of an edge, in meters per minute, of the cube when the volume is exactly 8 cubic meters.

Answer 1

Answer:

-1/4 meter per minute

Step-by-step explanation:

Since, the volume of a cube,

$V=r^3$

Where, r is the edge of the cube,

Differentiating with respect to t ( time )

$\frac{dV}{dt}=3r^2\frac{dr}{dt}$

Given, $\frac{dV}{dt}=-3\text{ cubic meters per minute}$

Also, V = 8 ⇒ r = ∛8 = 2,

By substituting the values,

$-3=3(2)^2 \frac{dr}{dt}$

$-3=12\frac{dr}{dt}$

$\implies \frac{dr}{dt}=-\frac{3}{12}=-\frac{1}{4}$

Hence, the rate of change of an edge is -1/4 meter per minute.