Question

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

Accepted Answer

Let the radius of semicircle be $r$ .

Let the base of the rectangle inscribed in the semicircle be $' x'$ and its height be $' y'$ .

Consider the $△ C E B$ ,

$C E^{2} = E B^{2} + B C^{2}$

$∴ r^{2} = {(\frac{x}{2})}^{2} + y^{2}$ $\Rightarrow y^{2} = r^{2} - {(\frac{x}{2})}^{2} . . . (1)$

Consider the area of the rectangle $A = x y$

$\Rightarrow A^{2} = x^{2} y^{2}$

Substituting $(1)$ in the above area equation, we get

$A^{2} = x^{2} [r^{2} - {(\frac{x}{2})}^{2}]$

$\Rightarrow Z = A^{2} = x^{2} r^{2} - \frac{x^{4}}{4} . . . (2)$

Differentiating the equation $(2)$ with respect to $x$ , we get

$\frac{d Z}{d x} = \frac{d (x^{2} r^{2} - \frac{x^{4}}{4})}{d x}$

$\Rightarrow \frac{d Z}{d x} = r^{2} (2 x) - \frac{4 x^{3}}{4}$

$\Rightarrow \frac{d Z}{d x} = 2 x r^{2} - x^{3} . . . (3)$

To find the critical point, we set

$\frac{d Z}{d x} = 0 \Rightarrow 2 x r^{2} - x^{3} = 0$

$\Rightarrow x (2 r^{2} - x^{2}) = 0$

$\Rightarrow x = 0$ or $(2 r^{2} - x^{2}) = 0$

$\Rightarrow x = 0$ or $x = r \sqrt{2}$

$∴ x = r \sqrt{2}$ as $x$ is length of the base of the rectangle

Consider differentiating the equation $(3)$ with $x$ , we get

$\frac{d^{2} Z}{d x^{2}} = 2 r^{2} - 3 x^{2} . . . (4)$

Now,

${[\frac{d^{2} Z}{d x^{2}}]}_{x = r \sqrt{2}} = 2 r^{2} - 3 (r \sqrt{2})^{2} = 2 r^{2} - 6 r^{2} = - 4 r^{2} < 0$

$\Rightarrow$ $Z$ is maximum at $x = r \sqrt{2}$ (By Second derivative test)

Substituting $x$ in equation $(1)$ , we get

$y^{2} = r^{2} - {(\frac{r \sqrt{2}}{2})}^{2} = r^{2} - \frac{r^{2}}{2} = \frac{r^{2}}{2}$

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

As the area of the inscribed rectangle is maximum for dimension $x = r \sqrt{2}$ and $y = \frac{r \sqrt{2}}{2}$ , its maximum area is given by

$A = (r \sqrt{2}) \times \frac{r \sqrt{2}}{2}$ $\Rightarrow A = r^{2}$

Hence, the maximum area of the rectangle inscribed inside a semicircle is $r^{2}$ square units.

R S Aggarwal Solutions for Exercise 6: EXERCISE

R S Aggarwal Mathematics Solutions for Exercise - R S Aggarwal Solutions for Exercise 6: EXERCISE

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