Question 62777: Could someone please help me with this. I just don't get algebra, but i'm getting better.
John has 300 feet of lumber to frame a rectangular patio (the perimeter of a rectangle is 2 times length plus 2 times width). He wants to maximize the area of his patio (area of a rectangle is length times width). What should the dimensions of the patio be, and show how the maximum area of the patio is calculated from the algebraic equation.
Show clearly the algebraic steps which prove your dimensions are the maximum area which can be obtained. Use the vertex formula to find the maximum area.
Found 2 solutions by stanbon, jai_kos:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! 2*length+2*width=300
Divide thru by 2 to get:
length+width = 150
Solve for width:
width=150-length
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Let the length be "x".
Then the width = "150-x"
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Area=(length)(width)
Area=x(150-x)
Area=-x^2+150x
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This is a quadratic with a=-1 and b=150
Maximum area occurs when x=-b/2a
x=-150/(-2)=75
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Dimensions are
width = 75 ft.
length = 150-75=75 ft.
The maximum area is 75^2=5625 sq ft.
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Cheers,
Stan H.
Answer by jai_kos(139) (Show Source):
You can put this solution on YOUR website! Let L be the length
Let W be the width
We know the perimeter is:
2(L + W) = 300
Solving for L:
L + W = 150
L = 150 - W
Now write a formula for the area:
Area = L * W
Area = (150 - W) * W
Area = -W² + 150W
You can see this is the formula for a parabola. I'll simplify things by expressing it in terms of x:
f(x) = -x² + 150x
But it isn't in vertex form:
f(x) = a(x - h) + k
To get it in that form, first pull out the -1 to get x²:
f(x) = -1(x² - 150x)
Now take the coefficient on the x term (-150), take half of it (-75) and square it (5625). Add and subtract this:
f(x) = -1(x² - 150x + 5625 - 5625)
Now you can write this as a square:
f(x) = -1[ (x - 75)(x - 75) - 5625)
Simplify:
f(x) = -1(x - 75)² + 5625
You have a downward facing parabola, with a maximum vertex at the point (h, k) or (75, 5625)
So the maximum is when the width is 75. When you solve for length you get that the length is also 75. So the maximum area for the patio is a square with sides of length 75 feet. The total area will be 5,625 sq. feet, a maximum.
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