ramadhar had 200m fencing wire to fence his field . what is the maximum area he can fence with one side of 50m.
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Answer:
Let area be
A
and the sides of rectangular field be
x
and
y
;
So,
A
=
x
⋅
y
Now, one side of the rectangle is already made with a fence.
There are 4 sides, two sides of
x
meters and two sides of
y
meters. Let a side of
y
meters be already fenced.
Then, the remaining three sides are to be fenced with a fence of
200
m
length,
So,
2
x
+
y
=
200
y
=
200
−
2
x
So,
A
=
x
(
200
−
2
x
)
A
=
200
x
−
2
x
2
For area to be maximum,
d
A
d
x
=
0
And,
d
2
A
d
x
2
<
0
Now,
d
A
d
x
=
200
−
4
x
200
−
4
x
=
0
x
=
50
Now,
d
2
A
d
x
2
=
−
4
Which is negative
Thus,
x
=
50
is a maximum.
Now,
2
x
+
y
=
200
2
(
50
)
+
y
=
200
y
=
100
Thus, the required area is:
A
=
x
⋅
y
=
50
⋅
100
=
5000
m
2
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