SOLUTION: A rain gutter is formed by bending up the sides by x inches of a 30-inch wide rectangular metal sheet as shown in the figure. (a) Find a function that models the cross-sectional a

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Question 515179: A rain gutter is formed by bending up the sides by x inches of a 30-inch wide rectangular metal sheet as shown in the figure.
(a) Find a function that models the cross-sectional area of the gutter in terms of x.
(b) Find the value of x that maximizes the cross-sectional area of the gutter.
(c) What is the maximum cross-sectional area for the gutter?
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
I don't see the figure to which the problem refers so I'll assume that the sides are bent up at right angles.
a) If the two sides measure x inches each and the sheet metal was 30 inches wide to begin with, then the cross-sectional area can be expressed by the function:
A+=+x%2830-2x%29
A+=+30x-2x%5E2
b) If you were to graph this function you would see a parabola that opens downward. The x-value at the maximum point (called the vertex) can be found by:
x+=+%28-b%29%2F2a where a = -2 and b = 30.
x+=+%28-30%29%2F2%28-2%29
x+=+7.5
c) The maximum cross-sectional area A%5Bm%5D is found by substituting this value of x into the function for the area.
A+=+30x-2x%5E2 Substitute x = 7.5
A%5Bm%5D+=+30%287.5%29-2%287.5%29%5E2 Evaluate.
A%5Bm%5D+=+225-112.5
A%5Bm%5D+=+112.5sq.in.