A closed cylindrical can of fixed volume V has radius r

[IntegrateMe]

(a) Find the surface area, S, as a function of r.

S = 2*pi*r^2 + 2*pi*r*h

I know how the 2pi(r)^2 is found, but where does the 2*pi*r*h come from?

(b) What happens to the value of S as r goes to infinity?

S also goes to infinity. As r increases, S increases.

(c) Sketch a graph of S against r, if V = 10 cm^3

This is where I'm having trouble. I don't even know where to begin. Any help?

 

[CompuChip]

I know how the 2pi(r)^2 is found

Where does it come from then? What part(s) of the cylinder haven't you accounted for?

(c) Sketch a graph of S against r, if V = 10 cm^3

This is where I'm having trouble. I don't even know where to begin. Any help?

You can express the volume V of the cylinder in terms of the radius r and height h. Then if you set V to 10, you can use this to derive an expression for h in terms of r (for example, if V would be V = h + r, then V = 10 would give you 10 = h + r, so h = 10 - r). You can plug that into the formula for S, which will leave you with only r as the variable.

(b) What happens to the value of S as r goes to infinity?

Now try this one again. Remember, the volume is constant, so if r increases, h must decrease. Using the relation between h and r you found, you can rewrite S so that it depends on r and V only, then you can properly take the limit.