Express the amount of material needed to make a can as a function of the radius. In other words, find a function A(r) such that A represents the surface area (in square inches) of the cylinder and r is the radius (in inches).

my answer (not sure)

A(r) = r^2 + 2r + 29 ?

This looks like a continuation of your last problem

Visualize taking a tin can apart.
you would have 2 circles, the top and the bottom
plus the sleeve that forms the can.
Pretend you are opening it up by cutting along the height of the can.
Would you not have a rectangle whose length is the circumference of the circle, and whose width is the height ?

So the total area would be
top + bottom + rectangle
= πr^2 + πr^2 + (2πr)h, where h is the height.
= 2πr^2 + (2πr)h

This is precisely what I used in the previous question.

You will need the height, or else be able to calculate it like in your previous question when the volume was given.

Thank you!!

To find the surface area of a cylinder, you need to consider both the lateral and base areas. The lateral area is the area of the curved surface, and the base area is the area of the top and bottom circles of the cylinder.

The lateral area of a cylinder is equal to the circumference of the base (2πr) multiplied by the height (which is the same as the height of the can). However, the problem only asks for the function in terms of the radius, so we won't consider the height for now. Hence, the lateral area is given by:

Lateral Area = 2πr

The base area of the cylinder is the area of a circle with radius r, which is given by:

Base Area = πr^2

To find the total surface area (A) of the cylinder, we need to add the lateral area and the base area:

A(r) = 2πr + πr^2

Simplifying further, we can factor out πr from both terms:

A(r) = πr(2 + r)

Therefore, the function A(r) represents the surface area of the cylinder in terms of the radius r:

A(r) = πr(2 + r)

Note that the constant 2 in the function comes from adding the lateral area to two times the base area, as there are two bases in a cylinder.