# Thread: Rectangular Sheet of Metal Problem

1. ## Rectangular Sheet of Metal Problem

A rectangular piece of sheet metal with an area of 1,600 in 2 is to be bent into a cylindrical length of stovepipe having a volume of 800 in 3. What are the dimensions of the sheet metal?

2. Hello, tmac11522!

A rectangular piece of sheet metal with an area of 1600 in²
is to be bent into a cylindrical length of stovepipe having a volume of 800 in³.
What are the dimensions of the sheet metal?
Code:
      *-----------*
|           |
W |           |
|           |
*-----------*
L

The area of this rectangle is: .$\displaystyle LW \:=\:1600 \quad\Rightarrow\quad W \:=\:\frac{1600}{L}$ .[1]
This is the height $\displaystyle h$ of the cylinder.

$\displaystyle L$ is the circumference of the circular base of the cylinder.
. . $\displaystyle 2\pi r \:=\:L \quad\Rightarrow\quad r \:=\:\frac{L}{2\pi}$ .[2]

The volume of the cylinder is 800: .$\displaystyle \pi r^2h \:=\:800$

Substitute [1] and [2]: .$\displaystyle \pi\left(\frac{L}{2\pi}\right)^2\left(\frac{1600}{ L}\right) \:=\: 800$

. . from which we get: .$\displaystyle L \:=\:2\pi$

Substitute into [1]: .$\displaystyle W \:=\:\frac{1600}{2\pi} \:=\:\frac{800}{\pi}$

Therefore, the dimensions are: .$\displaystyle 2\pi \times \frac{800}{\pi}$ inches.