Question

a large cube of ice is melting, each side shrinking by 8 centimeters per minute. find the rate at which the volume of this cube is changing at the moment when the length of an edge is 200 centimeters. assume that the cube maintains its shape.

Explanation:

Step1: Write the volume formula of a cube

The volume formula of a cube is \(V = s^{3}\), where \(s\) is the side - length of the cube.

Step2: Differentiate the volume formula with respect to time \(t\)

Using the chain rule, \(\frac{dV}{dt}=3s^{2}\frac{ds}{dt}\)

Step3: Substitute the given values

We are given that \(s = 200\) cm and \(\frac{ds}{dt}=- 8\) cm/min (negative because the side - length is shrinking).
Substitute these values into the formula \(\frac{dV}{dt}=3\times(200)^{2}\times(-8)\)
First, calculate \((200)^{2}=40000\)
Then, \(3\times40000\times(-8)=3\times(-320000)=-960000\)

Answer:

The rate at which the volume of the cube is changing is \(-960000\) cubic centimeters per minute.