Question

What dimensions of the rectangle will result in a cylinder of maximum volume if you consider a rectangle of perimeter 12 inches in which it forms a cylinder by revolving this rectangle about one of its edges?

Answer 1

A cylinder's volume, `V=pir^2h`
The width of the rectangle, which forms the circumference, `w=2pir`
graph{x+y=12 [-2.49, 25.99, -1.51, 12.74]}
The graph shows possible lengths for the sides of the rectangle if they must add up to 12 inches (`w` is horizontal and `h` is the vertical axis).
At what point along that curve is `w^2/(4pi^2)*h=r^2h` the greatest?

graph{(12x-x^2)/(4pi) [-0.89, 13.35, -1.027, 6.1]}

Since `r=w/(2pi)=(12-h)/(2pi)`, you can plot a graph of the volume for different values of `h` with the equation `y=pi*(12-x)/(4pi^2)*x=(12x-x^2)/(4pi)`. Differentiating to find the maximum:

`dy/dx=12/(4pi)-(2x)/(4pi)=3/pi-x/(2pi)=0`

`x=h=6`

Since `h=6`, `w` must also equal `6`

`V=pir^2*h=pi*(6/(2pi))^2*6=533pi=1674`