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 | |
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*-----------*
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.