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 feet.

The height of the shelter is feet.

The depth of the shelter is 4 feet.

The roof's area is ft².

At $5/ft², its cost is: dollars.

The sides have an area of ft².

At $2/ft², they will cost: dollars.

The total cost is: .

. . Solve for .**[1]**

The volume of the shelter is: . .**[2]**

Substitute [1] into [2]: .

Hence: . is the function you must maximize.