Aclosed 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.

To express the height as a function of the volume and the radius, we are going to use the volume formula for a cylinder: 
where
is the volume 
is the radius 
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: . We also know that the radius is r cm and height is h cm, so  and . Lets replace the values in our formula:






Next, we are going to use the formula for the area of a cylinder: 
where
is the area 
is the radius 
is the height

We know from our previous calculation that , so lets replace that value in our area formula:



By the commutative property of addition, we can conclude that:

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