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If three sides of a rectangular park have a total length $400$ ft, then the area of the park is maximum when the length (in ft) of its longer side is

  1. $299$
  2. $200$
  3. $201$
  4. $399$
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Let the one side be $L$ ft, and the other side be $B$ ft.

Let's draw the diagram for a better understanding.



Given that,  $L+2B=400$

$\Rightarrow 2B=400-L$

$\Rightarrow \boxed{B=\frac{(400-L)}{2}}$

Area of rectangle $A=L \ast B$

$\Rightarrow A=\frac{L\ast(400-L)}{2}$

$\Rightarrow A=200L-\frac{L^{2}}{2}$

The area is maximum when differentiation of $A$ is zero.

$\Rightarrow  200-\frac{2L}{2}=0$

$\Rightarrow 400-2L=0$

$\Rightarrow 2L=400$

$\Rightarrow \boxed{L=200\;\text{ft}}$

And, $B=\frac{400-200}{2}$

$\Rightarrow B = \frac{200}{2}$

$\Rightarrow\boxed{B = 100\;\text{ft}}$

$\therefore$ The length of the longer side is $200\;\text{ft}.$

Correct Answer $:\text{B}$

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