a metal cylinder is heated and then cools. if the radius always equals the height, find the expression for the instantaneous rate of change of volum V with respect to the radius r.
To find the expression for the instantaneous rate of change of volume with respect to the radius, we need to calculate the derivative of the volume function with respect to the radius.
Given that the radius always equals the height, we can express the volume of the cylinder as:
V = πr^2h
Since the radius and height are equal, we can substitute r for h:
V = πr^2r
V = πr^3
To find the rate of change with respect to the radius, we differentiate the volume formula with respect to the radius:
dV/dr = d(πr^3)/dr
Using the power rule for differentiation, where d(x^n)/dx = nx^(n-1), we can differentiate the formula for volume:
dV/dr = 3πr^2
Therefore, the expression for the instantaneous rate of change of volume with respect to the radius (dV/dr) is 3πr^2.