SOLUTION: making a rain gutter a rain gutter is formed by bending up the sides of a 30-inch-wide rectangular metal sheet (a) find a function that models the cross- sectional area of the gu

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Question 843192: making a rain gutter
a rain gutter is formed by bending up the sides of a 30-inch-wide rectangular metal sheet
(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
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the perimeter of a rectangle is equal to 2L + 2W.
the area of a rectangle is equal to LW.

the rectangle is the cross section of the gutter.

you know that the perimeter is equal to 30, so you set up an equation as shown below:

2L + 2W = 30

solve for L in terms of W.

you will get L = (30 - 2W) / 2 which simplifies to L = 15 - W

substitute for L in the equation of Area = LW to get:

Area = (15-W) * W

After you simplify, your equation becomes:

Area = 15W - W^2

rearrange the terms to get:

Area = -W^2 + 15W

this equation is now in the standard form of a quadratic equation which is equal to ax^2 + bx + c

a = -1
b = 15
c = 0

since the coefficient of the w^2 term in this quadratic is negative, the graph will point up and open down.

this means the max/min point of the quadratic equation is a max point.

the x-coordinate of the maximum point in the graph of this equation is equal to -b/2a which is equal to -15/-2 which is equal to 7.5.

the y-coordinate of the maximum point in the graph of this equation is equal f(7.5) which is equal to -(7.5)^2 + (15 * 7.5) which is equal to 56.25.

all you do is replace w in the original equation with 7.5 and solve for the area.

-w^2 + 15w becomes -(7.5)^2 + 15*7.5

the maximum area is at 56.25.

it occurs when W is equal to 7.5

When W is equal to 7.5, the perimeter of the gutter becomes 2L + 2(7.5) = 30 which simplifies to 2L + 15 = 30

solve for L to get L = 7.5.

the maximum cross sectional area is when both L and W are equal to 7.5.

the graph of the equation of the area of the cross section of the gutter is shown below:

graph%28600%2C600%2C-5%2C20%2C-10%2C60%2Cx%2A%2815-x%29%2C56.25%29

the formula that was used to generate this graph was:

y = (15 - x) * x

length is represented by (15 - x) and width is represented by x.

the horizontal line at y = 56.25 is there to show you what the top of the curve maxes out at 56.25.