Calculus differentiation applications

Question

I have to make an open cylindrical out of 48 square feet of material. That is, this can will not have a top. What are the dimensions of the radius and height that will maximize the volume of the can . Must be calculus work.

Answer 1

I will set this up for you and you can work the math.

 

Assuming the can has a bottom, but no top,

 

A = πr² + 2πrh = 48 ft² ⇒

 

h(r) = (48 - πr²) / (2πr)

 

V = πr²h = πr²h(r) = πr² [(48 - πr²) / (2πr)] = r(48 - πr²) / 2 = (1/2) (48r - πr³)

 

Now set dV/dr = 0 and solve for r, and then plug the value of r into the h(r) equation to obtain h.   The numbers will come out in feet. You should be able to finish from here.