SOLUTION: for 100 ft edging materials, what are the dimensions of the rectangle with the maximum area? what is the area? what is the area is changing the shape to a perfect circle?

Question 1073903: for 100 ft edging materials,
what are the dimensions of the rectangle with the maximum area?
what is the area?
what is the area is changing the shape to a perfect circle?

Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
let l be the length and w be the width, then
:
2l + 2w = 100
:
1) l + w = 50
:
2) Area(A) of rectangle = l * w
:
solve equation 1 for l and substitute for l in equation 2
:
l = 50 - w
:
A = (50 - w) * w = 50w - w^2
:
To find max area, we take the first derivative and set it equal to zero since A is the equation for a parabola that opens downward
:
A' = 50 - 2w
:
50 -2w = 0
:
w = 25 and l = 25
:
***********************************
max A = 25 * 25 = 625 square feet
***********************************
:
circumference(C) of a circle = pi * diameter(d)
:
100 = (22/7) * d
:
d = (100 * 7) / 22 = 31.8182
:
radius(r) = 31.8182 / 2 = 15.9091
:
***********************************************************************
Area of circle = pi * r^2 = (22/7) * (15.9091)^2 = 795.4555 square feet
***********************************************************************
:

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