A rectangular piece of paper is curled into a cylinder with two open circles on each side. The perimeter of the piece of paper is 124 inches. What is a function that could be written to find the length and width of the paper where the volume of the cylinder is at a maximum?
2026-04-13 12:01:26.1776081686
Writing the function to maximize volume or a cylinder
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Here are some hints.
The piece of paper is rectangular, so you have two legs for the length $L$ and two legs for the width $W$. Hence, the perimeter $P=2W+2L$.
You know the perimeter, so you know $2W+2L=124$ (inches). With this in mind you can solve for $L$ in terms of $W$, or vice versa.
This lets you express the volume of the cylinder in terms of $L$ only, or $W$ only. Following this, you can use that formula to find the maximum volume (I'd use a derivative) but it doesn't look like you're asked to go that far.
Can you take it from here?