Find the objective function for the given calculus problem. A rectangular yard i

Question

Find the objective function for the given calculus problem. A rectangular yard is to be enclosed with a fence by attaching it to a house whose length is 40 feet. See the figure below. The amount of fencing to be used is 240 feet. Find the dimensions of the yard so that the greatest area is enclosed. (Let A represent the area enclosed. Assume that the fence extends a distance x from the corner of the house and then wraps around to form a rectangle. Enter your answer in terms of x.) A(x) = Give the domain of the objective function, but do not actually attempt to solve the problem. (Enter your answer using interval notation.)

Explanation / Answer

Perimeter of yard,
P = 240 + 40 = 280 = 2(2x + 40) + 2a
where a is the length of the other sides of the yard.
4x + 80 + 2a = 280
2a = 280 - (4x + 80)
2a = 200 - 4x
a = 100 - 2x
so one side = 2x + 40 and the other side = 100 - 2x
A = (2x + 40)(100 - 2x)
A = -4x^2 + 120x + 4000...The domain is all real numbers.
find dA = 0 to determine the distance of x that maximizes A.

It is too tempting for me to determine the actual solution so
dA = -8x + 120
-8x + 120 = 0
x = 15
x = 15.
This means that area is optimized with one side being 30 + 40 = 70.
The other side then, is also 70
and the maximum area is 70^2 = 4900,

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