Question

A window is in the form of a rectangle surmounted by a semi-circle. If the total perimeter is 9metres , find the radius of the semi circle for the greatest window area.​

Answer 1

Question :-

• A window is in the form of a rectangle surmounted by a semi-circle. If the total perimeter is 9 metres, find the radius of the semi circle for the greatest window area.

Answer

Given :

• A window is in the form of a rectangle surmounted by a semi-circle.
• The total perimeter of window is 9 metres.

To find :-

• The radius of the semi circle for the greatest window area.

Concept used :-

• Application of Derivatives : Maxima and Minima.

Step I: Let f(x) be a differentiable function on a given interval and let f’’ be continuous at stationary point.

Find f'(x) and for maximum or minimum points, put the equation f' (x) = 0 and let x = a, b, … be solutions.

Step II: Case (i) : If f ‘‘ (a) < 0 then f(a) is maximum.

            Case (ii): If f ‘’(a) > 0 then f(a) is minimum.

Note:

(i) If f’’(a) = 0 the second derivatives test fails in that case we have to go back to the first derivative test.

(ii) If f’’(a) = 0 and a is not a point of local maximum nor local minimum then a is a point of inflection.

Solution :-

Let Length of rectangle be '2x' metres and breadth be 'y' metres.

So, radius of semicircle = 'x' metres.

★ Step - 1 :-

Perimeter of window = 9 metres

⇛ 2x + y + y +  πx = 9

⇛ x( π + 2) + 2y = 9

⇛2y = 9 - x( π + 2) ......[1]

★ Step - 2 :-

Area of window, A = Area of rectangle + Area of semicircle

⇛ A = 2xy + 1/2 π x²

On substituting the value of y, from [1] we get

⇛ A = f(x) = x[9 - x( π + 2)] + 1/2 π x²

⇛ f(x) = 9x - π x² - 2 x² + 1/2 π x²

⇛ f(x) = 9x - 2 x² - 1/2 π x²

★ Now, Differentiate w. r. t. x

⇛ f'(x) = 9 - 4x - πx

⇛ f'(x) = 9 - (4 + π)x ..............[2]

★ For maxima or minima

Put f'(x) = 0, we get

⇛ 9 - (4 + π)x = 0

⇛ (4 + π)x = 9

$\bf \:⇛x = \dfrac{9}{\pi \: + 4} \: metres$

★ Differentiate [2] w. r. t. x, we get

⇛ f''(x) = 0 - (4 + π)

⇛ f''(x) = - (4 + π)

⇛ f''(x) < 0

⇛ f(x) is maximum.

⇛ A is maximum.

⇛ Area of window is maximum.

$\bf \:So, \: radius \: of \: semicircle = \dfrac{9}{\pi \: + 4} \: metres$

Answer 2

Answer:

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