Activity 5.3.3.
Suppose that we want to construct a cylindrical can using \(60\) square inches of material for the surface of the can. In this context, how does the can’s volume depend on the radius we choose?
(a)
Use the formula for the surface area of a cylinder and the given constraint that the can’s surface area is \(60\) square inches to write an equation that connects the radius \(r\) and height \(h\text{.}\)
(b)
(c)
Recall that the volume of a cylinder is \(V = \pi r^2 h\text{.}\) Use your work in (b) to write \(V\) as a function of the single variable \(r\text{;}\) simplify the formula as much as possible.
(d)
What is the domain of the function \(V\) in the context of the physical setting of this problem? (Hint: how does the constraint on surface area provide an upper bound for the value of \(r\text{?}\) Think about the maximum area that can be allocated to the top and bottom of the can.)
