SOLUTION: Among all rectangles that have a 20 ft perimeter find the dimensions of the one with the largest area?

Question 278610: Among all rectangles that have a 20 ft perimeter find the dimensions of the one with the largest area?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

The perimeter of a rectangle is given by:

Solving for

The area of a rectangle is given by:

Substituting for

Putting the quadratic into standard form:

Hence

and

and the vertex of the concave down parabola is at

Therefore the maximum area is when the width is one-fourth of the perimeter. If the width is one-fourth of the perimeter, 2 times the width is one-half of the perimeter and then 2 times the length is also one-half of the perimeter (see the perimeter formula). And then the length must also be one-fourth of the perimeter. So the maximum area rectangle for ANY given perimeter is a square with a side measure one-fourth of the perimeter.

John

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