SOLUTION: using 24 feet, what is the largest rectangle area possible?
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Question 517998
:
using 24 feet, what is the largest rectangle area possible?
Answer by
solver91311(24713)
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Presuming you mean a maximum 24 foot perimeter:
The rectangle with the greatest area for a given perimeter is a square with sides that measure the perimeter divided by 4.
The perimeter of a rectangle is given by:
The area of a rectangle is given by:
Substituting from the perimeter equation:
This function graphs to a parabola opening downward meaning that the vertex is a maximum.
Algebra solution:
The vertex is the maximum, so for a parabola with equation
, the vertex is at
. For this parabola:
Calculus solution:
The maximum value of the function, hence the maximum area, is where the value of the first derivative is equal to zero:
Set equal to zero:
Hence, the maximum area rectangle for a given perimeter is a square with sides of length one-fourth of the perimeter.
John
My calculator said it, I believe it, that settles it
