Hello, doctorgk!

Did you make a sketch?

An open rectangular storage shelter is 4 ft. deep, consisting of

two vertical sides and a flat roof, is attached to an existing structure.

The flat roof is made of aluminum and costs 5 dollars per square foot.

The two sides are made of plywood costing 2 dollars per square foot.

If the budget for the project is $400, determine the dimensions

of the shelter which will maximize the volume. Code:

x
* - - - - - - *
4 / /|
/ / |
/ / |y
* - - - - - - * |
| | | |
| * | *
y| / | /
| / | / 4
|/ |/
* *

The width of the shelter is $\displaystyle x$ feet.

The height of the shelter is $\displaystyle y$ feet.

The depth of the shelter is 4 feet.

The roof's area is $\displaystyle 4x$ ft².

At $5/ft², its cost is: $\displaystyle 20x$ dollars.

The sides have an area of $\displaystyle 2(4y) = 8y$ ft².

At $2/ft², they will cost: $\displaystyle 2(8y) = 16y$ dollars.

The total cost is: .$\displaystyle 20x + 16y \:=\:400$

. . Solve for $\displaystyle y\!:\;\;y \:=\:25 - \frac{5}{4}x$ .**[1]**

The volume of the shelter is: .$\displaystyle V \;=\;4xy$ .**[2]**

Substitute [1] into [2]: .$\displaystyle V \;=\;4x\left(25 - \frac{5}{4}x\right)$

Hence: .$\displaystyle \boxed{V \;=\;100x - 5x^2}$ is the function you must maximize.