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

Answer
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Hint:First, evaluate the perimeter of the figure and equate it to 10. To admit maximum light through the whole opening we have to maximize the area of the figure to evaluate the dimensions of the window.

Complete step-by-step answer:
Draw the figure of the window.
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ABCD is the rectangle and O is the centre of the semi-circle.
We are given that a window is in the form of a rectangle surmounted by a semi-circle as shown in the figure.
First, we let the length and breadth of the rectangle.
Let x and y be the length and breadth of the rectangle respectively.
Since, the rectangle is surmounted by the semi-circle therefore, diameter of the semicircle is equal to the length of the rectangle.
Therefore, the diameter of the semicircle is x.
The radius is half of the diameter.
Therefore, the radius of the semicircle is x2.
The perimeter of the whole figure includes two-time breadth, one-time length and the perimeter of the semicircle.
We know that the perimeter of the semicircle of radius ris πr.
Write the perimeter of the whole figure and equate it to 10.
x+2y+πx2=10
Solve for y.
2x+4y+πx=204y=20πx2xy=20(π+2)x4...........(1)
Now we maximize the area of the window.
Let the area of the window is A which is equal to the sum of the area of the rectangle and area of the semicircle.
Area of the rectangle is the product of the length and the breadth.
Area of the semicircle of radius r is πr22.
Therefore,
A=xy×12π(x2)2
Substitute the value of y.
A=x[20(π+2)x4]+πx28A=20x(π+2)x24+πx28
To maximize the area, differentiate A with respect to x.
To maximize the area, differentiate with respect to .
Use ddx(xn)=nxn1
dAdx=202(π+2)x4+2πx8…………..(2)
Equate it to 0.
dAdx=0202(π+2)x4+2πx8=0202(π+2)x4+πx4=0202(π+2)x+πx=0202πx4x+πx=020πx4x=020x(π+4)=0x(π+4)=20x=20π+4
Differentiate equation (2)with respect to x
Use ddx(xn)=nxn1
d2Adx2=02(π+2)4+π4d2Adx2=02π4+π4d2Adx2=π4π+4d2Adx2=(π+4)π+4d2Adx2=1
Since, at x=20π+4the value of d2Adx2<0therefore, the value of A is maximum at x=20π+4.
Substitute the value of x=20π+4 in equation (1).
y=20(π+2)20π+44y=20π+8020π404(π+4)y=10π+4
Radius of the semicircle is half of side DC
x2=10π+4
Hence, the length and breadth of the rectangle is 20π+4 and 10π+4 respectively and radius of the semicircle is 10π+4.

Note:
In these types of questions evaluate the one variable in terms of another variable to maximize the terms.
We use a double derivative test to find maximum and minimum value of the function. If the value of double derivative at critical points of the function is positive then minimum and when negative then maximum value occurs of the function.
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