Question

if the volume of CYLINDER WITH A height of 3 feet is 75 pie cubic feet, find the surface area of the cylinder in square feet

Answer 1

first:
the area of the cylinder is based on 2 things: height and radius.
its the area of the circle, multiplied by the height.

\[ \pi r^2h\]

Answer 2

you can reverse this to find the radius:
\[ A = \pi r^2h \]
\[ \frac{A}{\pi h} = r^2\]
\[r=\sqrt{\frac{A}{\pi h}}\]

Answer 3

sorry that should be V in the equation above not A

Answer 4

that's for volume.

Answer 5

so \[ r = \sqrt{\frac{V}{\pi h}}\]

Answer 6

you can then use this information to find the surface area of the cylinder. it has the area of 2 circles at the top and bottom, and then the perimeter of a circle multiplied by the height.

\[ SA = 2*\pi r^2 + 2\pi rh\]
or
\[SA = 2\pi r(r + h)\]

Answer 7

after doing some algebra, you could go directly to the surface area:

\[SA = 2\pi\left(\sqrt{\frac{V}{\pi h}}\right)^2 + 2\pi h\sqrt{\frac{V}{\pi h}}\]
\[SA = \frac{2\cancel{\pi} V}{\cancel{\pi} h} + \sqrt{\frac{4\pi^{\cancel{2}}h^{\cancel{2}}V}{\cancel{\pi} \cancel{h}}}\]
and you're left with:
\[SA = \frac{2V}{h} + \sqrt{4\pi hV}\]

Answer 8

\[SA = \frac{2*75\pi}{3} + \sqrt{4\pi*3*75\pi}\]
\[SA = 50\pi + \sqrt{900\pi^2}\]
\[SA = 50\pi + 30\pi\]
\[SA = 80\pi\]