A rectangular garden has a perimeter of 120 feet. How do you find an equation for the area of the rectangle as a function of the width, then determine the length and width of the rectangle which provide the maximum area?
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Answer:
See below
Explanation:
We call x to width and y to high
If perimeter is 120, then
2
x
+
2
y
=
120
, or
y
=
60
−
x
The surface
s
(
x
)
=
x
⋅
y
=
x
(
60
−
x
)
is the surface area as function of width
If we want to maximize area, then
s
´
(
x
)
=
60
−
2
x
=
0
and we obtain as solution
x
=
30
and then the surface will be maximum if rectangle is a square with 30 ft of side
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