SOLUTION: Two hundred and forty meters of fencing is available to enclose a rectangular playground. What should be the dimensions of the playground to maximize the area?
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-> SOLUTION: Two hundred and forty meters of fencing is available to enclose a rectangular playground. What should be the dimensions of the playground to maximize the area?
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Question 221882: Two hundred and forty meters of fencing is available to enclose a rectangular playground. What should be the dimensions of the playground to maximize the area? Answer by solver91311(24713) (Show Source):
so the length of a rectangle in terms of perimeter and width is:
The area of a rectangle is given by:
Substituting we can create an Area function in terms of width for any given perimeter:
Putting this quadratic function in standard form results in:
This graphs to a parabola opening downward, hence the vertex is a maximum. The vertex of a parabola represented by has an -coordinate of , hence the vertex for our Area function is:
Hence, the width giving the maximum area is . But twice the width is then , hence for any given perimeter, twice the length must be as well. Therefore, for the maximum area, the length must be equal to the width and the maximum area shape is a square with each side measuring one-fourth of the perimeter.
