Hello, Ryan!

A gutter is to be made by folding up the edges of a strip of metal.

If the metal is 12 in wide and the cross-sectional area of the gutter is to be 16 7/8 in sq,

what are the width and depth of the gutter? Code:

This is the strip of metal before folding.
: - - - - -12 - - - - - :
*-----*-----------*-----*
: x : 12-2x : x :
This is the gutter after folding.
* *
| |
x | | x
| |
*-----------*
12-2x

The cross-sectional area is: .$\displaystyle A \;=\;x(12-2x)$

We're told that this area will be: $\displaystyle 16\frac{7}{8} \:=\:\frac{135}{8}$ inē.

There is our equation . . . $\displaystyle x(12-2x) \:=\:\frac{135}{8}$

Multiply by 8: .$\displaystyle 8x(12-2x) \:=\:135 \quad\Rightarrow\quad 16x^2 - 96x + 135 \:=\:0$

Factor: .$\displaystyle (4x-9)(4x-15) \:=\:0 \quad\Rightarrow\quad x \;=\;\frac{9}{4},\;\frac{15}{4}$

There are **two** solutions . . .

. . . $\displaystyle \begin{array}{|c|c|}\hline

\text{Height} & \text{Width} \\

x & 12-2x \\ \hline \hline

2.25 & 7.5 \\ \hline \hline

3.75 & 4.5 \\ \hline

\end{array}$