the edges of a cube are increasing at the rate of 5 in/min. How fast is the volume changing when each side is 4 in.
To find the rate at which the volume is changing, we need to differentiate the volume of the cube with respect to time.
Let's denote the volume of the cube as V and the side length of the cube as s. We are given that ds/dt = 5 in/min. We want to find dV/dt when s = 4 in.
The volume of a cube is given by the formula V = s^3. So, we can express the volume as a function of s:
V = s^3
To differentiate both sides of the equation with respect to time t, we can use the chain rule:
dV/dt = d(s^3)/dt
Applying the chain rule, we get:
dV/dt = 3s^2 * ds/dt
Now, we can substitute the given values: s = 4 in and ds/dt = 5 in/min into the equation to calculate the volume's rate of change:
dV/dt = 3(4^2) * 5
dV/dt = 3(16) * 5
dV/dt = 240 in^3/min
Therefore, when each side of the cube is 4 inches, the volume is changing at a rate of 240 cubic inches per minute.