plug in x = 16 to get . Because the value of the second derivative is
positive, according to the second derivative test (see this page), the perimeter is a minimum at
x = 16.
- Radius is inches.
First, let’s draw a picture.
The volume of a cylinder is V = πr^2 h = 512. The material for the can is the surface area of the
cylinder (don’t forget the ends!) S = 2πrh + 2πr^2 . If we solve the volume equation for h, we get
h = . Now we can substitute this for h in the surface area equation: S = 2πr + 2πr^2
= + 2πr^2 . Now we take the derivative of S: + 4πr. If we solve this for r,
we get r = inches. We can verify that this is a minimum by taking the second derivative:
+ 4π. Next, we plug in and we can see that the value of the second
