Activity 5.4.5.
Suppose that we want to build an open rectangular box (that is, without a top) that holds \(15\) cubic feet of volume. If we want one side of the base to be twice as long as the other, how does the amount of material required depend on the shorter side of the base? We investigate this question through the following sequence of prompts.
(a)
Draw a labeled picture of the box. Let \(x\) represent the shorter side of the base and \(h\) the height of the box. What is the length of the longer side of the base in terms of \(x\text{?}\)
(b)
Use the given volume constraint to write an equation that relates \(x\) and \(h\text{,}\) and solve the equation for \(h\) in terms of \(x\text{.}\)
(c)
Determine a formula for the surface area, \(S\text{,}\) of the box in terms of \(x\) and \(h\text{.}\)
(d)
Using the constraint equation from (b) together with your work in (c), write surface area, \(S\text{,}\) as a function of the single variable \(x\text{.}\)
(e)
What type of function is \(S\text{?}\) What is its domain?
(f)
Plot the function \(S\) using Desmos. What appears to be the least amount of material that can be used to construct the desired box that holds \(15\) cubic feet of volume?
