# optimization

• Jan 31st 2008, 05:27 PM
doctorgk
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
• Jan 31st 2008, 07:20 PM
Soroban
Hello, doctorgk!

Did you make a sketch?

Quote:

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.