Math, asked by priyad1509, 1 month ago

Mr Shashi, who is an architect, designs a building for a small company. The design of window on the ground floor is proposed to be different than other floors. The window is in the shape of a rectangle which is surmounted by a semi-circular opening. This window is having a perimeter of 10 m as shown below Based on the above information answer the following: (i) If 2x and 2y represent the length and the breadth of the rectangular portion of the window, then the relation between the variables is given by (a) 4y - 2x =10 − π (b) 4y =10 -(2 − π)x (c) 4y= 10- (2 + π)x (d) 4y -2x =10 + π (ii) The combined area (A) of the rectangular region and semi-circular region of the window expressed as a function of x is (a) A =10x+( 2+ 1 2 π) 2 (b) A =10x-( 2+ 1 2 π) 2 (c) A =10x-( 2- 1 2 π) 2 (d) A= 4xy+ 1 2 π 2 (iii) The maximum value of area of the whole window, A is (a) A= 50 4+ 2 (b) A= 50 4+ 2 (c) A= 100 4+ 2 (d) A= 50 4− 2 (iv) The owner of this small company is interested in maximizing the area of the whole window so that maximum light input is possible. For this to happen, the length of rectangular portion of the window should be () 20 4+ m (b) 10 4+ m () 4 10+ m (d) 100 4+ m (v) In order to get the maximum light input through the whole window, the area (in sq. m) of the semi-circular opening of the window is (a) 100 (4+) 2 (b) 50 4+ (c) 50 (+4) 2 (d) same as the area of rectangular portion of the window


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Answers

Answered by shkulsum3
1

The perimeter of the window is given as 10m.

Since the window is in the shape of a rectangle surmounted by a semi-circular opening, we can divide the perimeter into two parts, one for the rectangle and one for the semi-circular opening.

The perimeter of a rectangle is given by 2(length + breadth). The perimeter of a semi-circular opening is given by π * diameter.

Let 2x and 2y represent the length and breadth of the rectangular portion of the window.

So, 2x + 2y = 10 - πd.

where d is diameter of the semi-circular opening.

Therefore, x + y = 5 - (π/2)d.

Now, we know that x = y + d, so substituting it into the first equation we get,

y + d = 5 - (π/2)d

y = 5 - (π + 2)d/2

On simplification, we get

y = (5 - π/2) - d

y = (10 - π) / 2 - x

So the relation between the variables is given by (c) 4y = 10 - (2 + π)x

(ii) The combined area of the rectangular region and semi-circular region of the window is given by

A = xy + (π/4)d^2

As we know that y = (10 - π) / 2 - x and d = x

A = x((10 - π) / 2 - x) + (π/4)x^2

On simplifying, we get

A = 10x - (π/2)x^2- (1/2)x^2 + (π/4)x^2

A = 10x - (π/4)x^2 - (1/4)x^2

A = 10x - (π/4 + 1/4)x^2

A = 10x - (x^2/4) (2+ 1/π)

So the combined area (A) of the rectangular region and semi-circular region of the window expressed as a function of x is

(a) A =10x+( 2+ 1 2 \pi) 2

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