When John decided to renovate his garage, he also wanted to attach it to his hou
ID: 3103603 • Letter: W
Question
When John decided to renovate his garage, he also wanted to attach it to his house. The only 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 68 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 easier. 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.
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Explanation / Answer
Formula:
let L = length
let w = width
2L + W = 68 [subtract 2L from both sides]
W = 68 - 2L
A = L x W
A = L x ( 68 - 2L)
A = 68L - 2L2
Choose two dimensions to plug into your formula.
Let L = 1
A = 68 x 1 - 2 x 12
A = 68 - 2 = 66
Next, use the vertex formula to find the dimension that produces the maximum area of the garage.
A = -2L2 + 68
a = -2, b = 68
Vertex Formula:
L= -b/2a = -68/(2x-2) = -68/-4 = 17ft
Finally, find the maximum area.
A = 68 x 17 - 2 x 172
A = 1156 - 578 = 578ft
Will the garage be large enough to accommodate the two cars?
Yes because the dimensions of the garage will be 17 x 34 (L x W) and that's certainly big enough to accommodate two big cars.
How does the length of the kitchen wall affect the structure of the garage?
The length of a kitchen wall will make a varaible into a constant and therefore limit the area of the garage.
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