optimization calculus

Question

Find the exact value of the maximum volume of a cylinder if the sum of the radius of the base and the height equal to 24cm.

Answer 1

You know that V = pi * r^2 * h. (Equation 1)

You're also told that r + h = 24. (Equation 2)

Solving Equation 2 for h = 24-r, you then plug that into Equation 1 to get:

V = pi * r^2 * (24-r) (Equation 3)

Now find dV/dr by taking the derivative of Equation 3.

You should get dV/dr = 48* pi* r - 3 *pi * r^2 (Equation 4)

The max occurs when dV/dr = 0.

Set Equation 4 equal to 0 and solve for r - this gives you the value of r at which the max volume occurs.

Then plug that r value back into Equation 3 to find the max volume of 6434.

You can check your answer on a graphing calculator by graphing Equation 3 and using the Calc feature to find the max