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Answer:
Therefore the length and width of the playground are 15.5 feet and 12.9 feet respectively.
Step-by-step explanation:
Given that, a homeowner plans to enclose a 200 square foot rectangle playground.
Let the width of the playground be y and the length of the playground be x which is the side along the boundary.
The perimeter of the playground is = 2(length +width)
=2(x+y) foot
The material costs $1 per foot.
Therefore total cost to give boundary of the play ground
=$[ 2(x+y)×1]
=$[2(x+y)]
But the neighbor will play one third of the side x foot.
So the neighbor will play
Now homeowner's total cost for the material is
where C(x) is total cost of material in $.
Given that the area of the playground is 200.
We know that,
The area of a rectangle is =length×width
=xy square foot
∴xy=200
Putting the value of y in C(x)
The domain of C is.
Differentiating with respect to x
Again differentiating with respect to x
To find the critical point set C'(x)=0
Therefore
Therefore at x= 15.5 , C(x) is minimum.
Putting the value of x in we get
=12.9
Therefore the length and width of the playground are 15.5 feet and 12.9 feet respectively.