Question

A closed can, in a shape of a circular, is to contain 500cm^3 of liquid when full. The cylinder, radius r cm and height h cm, is. made from thin sheet metal. The total external surface area of the cylinder is A cm^2.
Show that A=2πr^2 + 1000/r.

Answer 1

To express the height as a function of the volume and the radius, we are going to use the volume formula for a cylinder: $V= \pi r^2h$
where
$V$ is the volume 
$r$ is the radius 
$h$ is the height 

We know for our problem that the cylindrical can is to contain 500cm^3 when full, so the volume of our cylinder is 500cm^3. In other words: $V=500cm^3$. We also know that the radius is r cm and height is h cm, so $r=rcm$ and $h=hcm$. Lets replace the values in our formula:
$V= \pi r^2h$
$500cm^3= \pi (rcm^2)(hcm)$
$500cm^3=h \pi r^2cm^3$
$h= \frac{500cm^3}{ \pi r^2cm^3}$
$h= \frac{500}{ \pi r^2}$

Next, we are going to use the formula for the area of a cylinder: $A=2 \pi rh+2 \pi r^2$
where
$A$ is the area 
$r$ is the radius 
$h$ is the height

We know from our previous calculation that $h= \frac{500}{ \pi r^2}$, so lets replace that value in our area formula:
$A=2 \pi rh+2 \pi r^2$
$A=2 \pi r(\frac{500}{ \pi r^2})+2 \pi r^2$
$A= \frac{1000}{r} +2 \pi r^2$
By the commutative property of addition, we can conclude that:
$A=2 \pi r^2+\frac{1000}{r}$