1. ## optimization

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 dollars, determine the dimensions of the shelter which will maximize the volume.

thanks

2. 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.