a We are asked to substitute S=1100 into S=π r^2+π rl and rearrange the resulting equation to make l the subject of the formula. Let's follow this instruction.
c To find the value of r that maximizes the volume of the hut, let's use the calculator to graph the function we found in Part B. We begin by pushing the Y= button and typing the equation of V in the first row. Note that we use x instead of r in the expression, because that is what the calculator understands.
To see the graph you will need to adjust the window.
For r=20 the area of the two half circles is already greater than the given surface area, so let's use 0 and 20 as horizontal bounds for the graph.
Push WINDOW, change the settings, and push GRAPH.
To find the maximum point on the graph push 2nd and TRACE, then choose maximum from the menu.
The calculator will prompt you to choose a left and right bound and to provide the calculator with a best guess of where the maximum might be.
Remember, we used x instead of r to enter the expression in the calculator.
The result we got means that r≈ 10.8 feet maximizes the volume of the hut.
Extra
Greatest possible volume of the hut.
The second coordinate of the maximum point also tells us that the maximum volume of the hut is approximately 3961 cubic feet.
