rate of change in volume

Question

The volume of a cube decreases at a rate of 10 meters per second. Find the rate at which the side of the cube changes when the side of the cube is 1 meters.

Answer 1

The problem can be solved using implicit differentiation. Let s = the length of a side of the cube.

The volume of the cube is: V = s^3. The rate of change in volume is dV/dt. The question is asking for the rate that the side length is changing. That would be ds/dt.

If we take the derivative of both sides of V = s^3 with respect to time, we get dV/dt = d/dt[s^3].

dV/dt = 3s^2(ds/dt)

From the problem, dV/dt = -10 s = 1 ds/dt = ?

Plug the numbers in and solve for ds/dt.

Answer 2

Let x be a side of the cube.

Volume of a cube, V = x³

Taking the derivative of V

V' = 3x²x' and V' = 10 m³/s

x' = V'/(3x²)

When x = 1

x' = (10)/(3(1)²) = 10/3 m/s decreasing

Answer 3