The largest area that can be enclosed by a fence of 240 m and a wall is 7200m²
It is given that the rectangular area will be fenced up against a long rectangular wall. Therefore, the fencing will be done on just three sides.
Let one side be x
Then the side opposite to it must be x cm as well
The sum of the three sides should be equal to 240 m
Hence the third side must be 240 - 2x m
Therefore the area of the region fenced must be
A = x(240 - 2x)
or, A = 240x - 2x²
Now we need to maximize the above equation
Taking the first derivative zero we get
A' = 0
or, 240 - 4x = 0
or, 60 - x = 0
or, x = 60
Now we will test whether the above output is maximum or not. The second derivative of A is
A'' = -4
Since A'' is a constant function,
Now, A''(60) = -4
Therefore, the maximum side will have lengths of 60 m and 120 m.
Hence the largest area that can be enclosed with the fence is
60 X 120 m²
= 7200 m²
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