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 6 cubic meters per minute, find the rate of change of an edge, in meters per minute, of the cube when the volume is exactly 64 cubic meters.

a) 6
b) 3/8
c) -1/8
d) -8
e) -48

Answer 1

The volume of a cube is calculated by cubing the edge length. To find the rate of change of the edge length, we can set up an equation and differentiate it with respect to time. Substituting the volume value, we can find the rate of change.

Step-by-step explanation:

The volume of a cube is calculated by cubing the edge length. Let's assume that the edge length of the larger cube is x meters. So, the volume of the larger cube is x^3 cubic meters. The rate at which the volume decreases is given as 6 cubic meters per minute. When the volume is exactly 64 cubic meters, we can set up an equation as follows:

x^3 - 6t = 64

where t is the time in minutes.

To find the rate of change of the edge length, we can differentiate both sides of the equation with respect to t:

3x^2(dx/dt) - 6 = 0

Simplifying, we get:

3(dx/dt) = 6/x^2

(dx/dt) = 6/(3x^2)

Substituting x = 4 (since the volume is 64), we get:

(dx/dt) = 6/(3(4^2)) = 6/48 = 1/8 meters per minute

Therefore, the rate of change of the edge length is 1/8 meters per minute.