# A rectangle is inscribed in a semi-circle of radius r with one of its sides on diameter of semi-circle.

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A rectangle is inscribed in a semi-circle of radius r with one of its sides on diameter of semi-circle. Find the dimensions of the rectangle so that its area is maximum. Find the area also.

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Let ABCD be the rectangle which is inscribed in a semi - circle with centre O and radius r.

Assume that the length of rectangle is 2x and breadth is 2y.

⇒ CD = AB = 2x

In right angle ΔACD,We get

x2 + 4y2 = r...(i)

(By Pythagoras theorem)

Now area of rectangle ABCD is given by,

⇒  $A=2x\times2y=4xy$

Let Z = A2 = 4(r2x2 - x4) ...(ii)

Then Z is maximum or minimum accordings as A is maximum or minimum.

Differentiating equation (ii) with respect to x, we get

$\frac{dZ}{dx}=$ 4[r2 . 2x - 4x3]

For maximum or minimum value of Z, we have

$\frac{dZ}{dx}=0$

⇒  r2 - 2x2 = 0

( ∵ x cannot be zero)

⇒  $x^2=\frac{r^2}{2}$

Now,

Thus area will be maximum when,

$x=\frac{r}{\sqrt 2}$

⇒  $2x =\sqrt 2r,$

Putting $x=\frac{r}{\sqrt 2}$ in equation (i), we obtain,

$y=\frac{r}{2\sqrt 2}$

⇒ 2y = $\frac{r}{\sqrt 2}$

Now,

A = $2x \times 2y$

= $\sqrt2r\,\times\,\frac{r}{\sqrt2}$

= r

Hence, the dimensions are $\sqrt2r$ and $\frac{r}{\sqrt2}$ and area = r2 sq units.