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The maximum area of a rectangle that can be formed with a fixed perimeter of 20 units is sq units
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1891 Upvotes
Verified Answer
The correct answer is:
25
Let the sides of rectangle is $x$ and $y$.
$$
\text { Given, } \begin{aligned}
20 & =2(x+y) \Rightarrow x+y=10 \\
A & =x y=x(10-x)=10 x-x^2 \\
\frac{d A}{d x} & =10-2 x \\
\frac{d A}{d x} & =0 \Rightarrow 10-2 x=0 \\
\Rightarrow \quad x & =5 \Rightarrow \frac{d^2 A}{d x^2}=-2 < 0
\end{aligned}
$$
$\therefore \quad$ Area of rectangle is maximum when $x=y=5$
$\therefore \quad$ Area of rectangle $=(5)^2=25$
$$
\text { Given, } \begin{aligned}
20 & =2(x+y) \Rightarrow x+y=10 \\
A & =x y=x(10-x)=10 x-x^2 \\
\frac{d A}{d x} & =10-2 x \\
\frac{d A}{d x} & =0 \Rightarrow 10-2 x=0 \\
\Rightarrow \quad x & =5 \Rightarrow \frac{d^2 A}{d x^2}=-2 < 0
\end{aligned}
$$
$\therefore \quad$ Area of rectangle is maximum when $x=y=5$
$\therefore \quad$ Area of rectangle $=(5)^2=25$
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