Question

Find the maximum area of a rectangle inscribed in a circle of radius r

Answer 1

Answer:

The maximum area of a rectangle inscribed in a circle of radius 'r' is: 2r².

Step-by-step explanation:

In the figure, we can see that the radius of the given circle is 'r' and the rectangle inscribed in it has a length of 'l' and breadth 'b'.

Thus, the diagonal of the rectangle is of length 2r.

We know, area of a rectangle is Length * Breadth.

Thus, the area of the given rectangle (A) = l*b

From the Pythogoras theorem, we have,

(2r)² = l² + b²

⇒ b² = 4r² - l²

⇒ b = √4r²- l²

Thus, A = l * √4r²- l²

Differentiating the Area (A) with respect to length 'l', we have,

$\frac{dA}{dl} = \frac{-2l^{2}}{2\sqrt{4r^{2}-l^{2} }} + \sqrt{4r^{2}-l^{2} }$

Since the Area of the rectangle is a constant term, we can write,

$\frac{dA}{dl} = 0$

⇒$\frac{-2l^{2}}{2\sqrt{4r^{2}-l^{2} } } + \sqrt{4r^{2}-l^{2}} = 0$

⇒ -2l² + 2 (4r²-l²) = 0

⇒ -2l² + 8r² - 2l² = 0

⇒ 8r² = 4l²

⇒ 2r² = l²

⇒$l=\sqrt{2r^{2} }$

⇒$l = \sqrt{2} r$

∵ b = √4r²- l²

∴ $b= \sqrt{4r^{2} - 2r^{2} } = \sqrt{2r^{2} } =\sqrt{2}r$

Hence, A = 2r².

Moreover, we observe that the length and breadth of the rectangle are same, and hence, the largest rectangle that can be inscribed in a circle, is a SQUARE.

Answer 2

Answer:

2r^2

Hope this will help you

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