Question #53d0e

1 Answer
Jan 2, 2018

Cylinder with radius of 4.3 cm and a height of 8.6 cm.

Explanation:

First, let's recognize that we must find the dimensions of a cylinder that yields a volume of 500 cm^3 and has the least surface area possible. This means that we must minimize the quantity 2pir^2 + 2pirh.

Since the volume of a cylinder is pir^2h, we have

pir^2h = 500.

h = 500/(pir^2).

Plugging back into our equation for surface area yields

A(r) = 2pir^2 + 1000/r.

Since we want to find the minimum of this function, we must take the derivative with respect to r and set the derivative equal to 0.

0 = 4pir + -1000/r^2.

Now, all we have to do to solve for r. Rearranging yields

1000/r^2 = 4pir

1000 = 4pir^3

r^3 = 250/pi

r = root(3)(250/pi).

Solving for h with this value of r yields

h = 500/(piroot(3)(250/pi)^2

h = 500/(pi(250^(2/3)/pi^(2/3)))

h = (2times250^(3/3))/(250^(2/3)times(pi^(3/3)/pi^(2/3)))

h = 2times250^(1/3)/pi^(1/3)

h = 2root(3)(250/pi).

So our final answer is a cylinder with a radius of 4.3 cm and height of 8.6 cm.