Name: Local Maxima and Minima of a Function
Uploaded: 2019-07-19
Duration: 14 min 17 s
Description: Do you know what is minima or maxima of a function? This video also explains how to find local maxima and minima by differentiation in a detailed manner.

Question

A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is

10 m

. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.

Accepted Answer

Let the dimensions of the rectangular part be $x$ and $y$ .

Radius of semi-circle $= \frac{x}{2}$

Total perimeter $= 10$

$\Rightarrow (x + 2 y) + π (\frac{x}{2}) = 10$

$\Rightarrow 2 y = [10 - x - π (\frac{x}{2})]$

$\Rightarrow y = \frac{1}{2} [10 - x (1 + \frac{π}{2})] . . . (1)$

Now,

Area, $A = \frac{π}{2} {(\frac{x}{2})}^{2} + xy$

$\Rightarrow A = \frac{{πx}^{2}}{8} + \frac{x}{2} [10 - x (1 + \frac{π}{2})]$ [ From eq. $(1)$ ]

$\Rightarrow A = \frac{{πx}^{2}}{8} + \frac{10 x}{2} - \frac{x^{2}}{2} (1 + \frac{π}{2})$

$\Rightarrow \frac{dA}{dx} = \frac{πx}{4} + \frac{10}{2} - \frac{2 x}{2} (1 + \frac{π}{2})$

For maximum or minimum values of $A$ , we must have

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

$\Rightarrow \frac{πx}{4} + \frac{10}{2} - \frac{2 x}{2} (1 + \frac{π}{2}) = 0$

$\Rightarrow x [\frac{π}{4} - 1 - \frac{π}{2}] = - 5$

$\Rightarrow x = \frac{- 5}{(\frac{- 4 - π}{4})}$

$\Rightarrow x = \frac{20}{(π + 4)}$

Substituting the value of $x$ in eq. $(1)$ , we get

$y = \frac{1}{2} [10 - (\frac{20}{π + 4}) (1 + \frac{π}{2})]$

$\Rightarrow y = \frac{5 π + 20 - 5 π - 10}{(π + 4)}$

$\Rightarrow y = \frac{10}{(π + 4)}$

$\frac{d^{2} A}{{dx}^{2}} = \frac{π}{4} - \frac{π}{2} - 1$

$\Rightarrow \frac{d^{2} A}{{dx}^{2}} = \frac{π - 2 π - 4}{4}$

$\Rightarrow \frac{d^{2} A}{{dx}^{2}} = \frac{- π - 4}{4} < 0$

Thus, the area is maximum when $x = \frac{20}{π + 4}$ and $y = \frac{10}{π + 4}$ .

So, the required dimensions are given below:

Length $= \frac{20}{π + 4} m$

Breadth $= \frac{10}{π + 4} m$

A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is $10 m$ . Find the dimension of the rectangular of the window to admit maximum light through the whole opening.

Important Questions on Maxima and Minima

Learn and Prepare for any exam you want !

A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.

Important Questions on Maxima and Minima

Learn and Prepare for any exam you want !

A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is $10 m$ . Find the dimension of the rectangular of the window to admit maximum light through the whole opening.