if the number of square unites in the surface area of a cylinder with a radius of 6 is 1/2 of the number of cubic units in the cylinder, what is the number of square units in the surface area?

Surface area... What's a cylinder look like? Top and bottom circles and the long part in the middle. What's the surface area of that? The outer surface. Pretend you have wrapping paper and you wrap it around the cylinder and seal it into place so that it adheres to the cylinder perfectly. Now take a pair of scissors and cut a STRAIGHT LINE up the length of the cylinder. Unravel the wrapping paper. The area of the wrapping paper is that part of the surface area.

What does it look like? A rectangle! The up-and-down part is the length of the cylinder going up and down. The right-to-left part is the distance from the place you made the cut on the cylinder... all the way around and back to that point. What is this otherwise known as? The circumference of the top (or bottom) circle!

So the area of the long part is the circumference of the circle (2pi*r) multiplied by the height of the cylinder (h) = 2pi*r*h.

The area of the two circles (top/bottom) = 2*pi*r^2.

So the total surface area is 2pi*r*h + 2pi*r^2.

The radius is 6 so you have 12pi*h + 72pi = 12pi(h + 6) for your surface area.

This is 1/2 the cubic units of the volume, so the volume is 24pi(h+6).

Volume = pi*r^2*h

We know r = 6 so volume = 36pi*h

So we have 24pi(h+6) = 36pi*h

24pi*h + 144pi = 36pi*h

144pi = 12pi*h

h = 12

Surface area was 12pi(h+6) so we have 12pi(18) = 216pi.

Check our work...

Surface area = 2pi*r*h + 2pi*r^2 = 144pi + 72pi = 216pi

Volume = pi*r^2*h = 432pi

Volume is 2x the surface area.

We're good.