Can someone help me with this math problem?

Question

When John, decided to renovate his garage, he also wanted to attach it to his house. The only one wall that was available to achieve this was the kitchen wall. 

The contractor informed him that the area of his new garage will be limited by the cost of the materials he should buy. After some calculations, they decided that he could afford three walls with a total length of 66 feet. The kitchen wall would serve as the fourth wall.

But John had two big cars, so he wanted to build a garage with the biggest possible area under his contractor’s constraints.

•What is the formula for the area of the garage? 
◦Hint: Use the formula for the perimeter of a rectangle and the lengths of the three sides of the garage, using W for width and L for length. Solve for W or L, whichever is easiest. Then substitute this expression into the Area formula (A=LW) so there are only two variables, A and one of the sides. 
•Choose two dimensions to plug into your formula.  
◦What areas result? If you get a negative area, your dimension was too large, so select another. 
◦Next, use the vertex formula to find the dimension that produces the maximum area of the garage. 
◦Finally, find the maximum area. 
•Will the garage be large enough to accommodate the two cars?
•How does the length of the kitchen wall affect the structure of the garage?
•Summarize your findings in writing using proper style and grammar.

vekkus4 · Answer

Which part of it do you need help with?
It is sort of trying to help you already by breaking the problem down into steps.
The first thing to do is:
 * Read it carefully

There are some weird things in the story to deal with ...
 * Figure out which parts are unnecessary, for example
 ... the garage belongs to John
 ... the wall he will use is on the kitchen
 ... John has two cars
 ... John talked to a contractor
 * Figure out what aspects of an actual real garage are missing from the statement just to make the math simpler:
  ... there seems to be no plan for the roof
 .... or a door on the front 
 ... or windows 
 ... the length of the kitchen wall doesn't appear to be known
 ... Then humor your poor teacher by pretending that you do not notice these things

Now state the problem more essentially

What's the maximum size rectangular enclosure you can build by adding three walls along an existing wall, given the new walls will add up to 66 feet?

Now for the Steps and Hints
Two variables can be W and L
The area is A = WL
The perimeter is going to be W + L + W = 66 (it could be L + W + L = 66 too but you will get the same shape either way, with the letters switched.)

Then you do A = W (66 - 2 W) = 66 W - 2 W^2   since you can solve for L and plug that in.

It is -2 ((W - 33/2)^2  - 1089/4)  completing the square so the focus is at W=33/2.
Also this is gotten using Calculus by finding dA/dW = 66 - 4 W so the critical point is at 66/4=33/2 or 16.5.

For "Summarize your findings" this is more of a writing exercise, not a math problem.  Try to imagine you are documenting the project described in the problem, mentioning John, the contractor, the walls, the kitchen, and the cars.

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Can someone help me with this math problem?

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