SOLUTION: A family wants to fence a rectangular play area alongside the wall of their house. The wall of their house bounds one side of the play area. If they want the play area to be exactl
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Question 1168282: A family wants to fence a rectangular play area alongside the wall of their house. The wall of their house bounds one side of the play area. If they want the play area to be exactly 2025ft^2, what is the least amount of fencing needed to make this? Round your answer to the nearest two decimal places Answer by ikleyn(52803) (Show Source):
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A family wants to fence a rectangular play area alongside the wall of their house.
The wall of their house bounds one side of the play area. If they want the play area to be exactly 2025ft^2,
what is the least amount of fencing needed to make this? Round your answer to the nearest two decimal places
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It is a Calculus problem.
Let x be the length of the side of the rectangle parallel to the wall,
and let y be the length of the perpendicular side.
They want you minimize the function f(x,y) = x + 2y under the restriction xy = 2025 ft^2.
You express y = and substitute it into the formula for f(x,y).
In this way, you come to the function g(x) = x + , and your task now is
to find the minimum value of this function.
Take the derivative g(x) over x and equate it to zero
g'(x) = 1 - = 0.
From this equation
x^2 = 4050,
x = = 63.64 ft.
It is the dimension of the rectangle along the wall.
The perpendicular dimension is y = = 31.82 ft.
ANSWER. The dimensions are 63.64 ft along the wall and 31.82 perpendicular to the wall.
The total length of the fencing is 63.64 + 2*31.82 = 127.28 ft.
