A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 metres. Find the dimensions of the window to admit maximum light through the whole opening. How having large windows help us in saving electricity and conserving environment ?
Answers
Toolbox:
Area of a circle =πr2
Area of a rectangle =lb
ddx(xn)=nxn−1
Step 1:
Perimeter of the window when the width of window is x and 2r is the length.
⇒2x+2r+12×2πr=10[Given]
2x+2r+πr=10
2x+r(2+π)=10------(1)
For admitting the maximum light through the opening the area of the window must be maximum.
A=Sum of areas of rectangle and semi-circle.
Step 2:
Area of circle=πr2
Area of rectangle=l×b=2×r×x
A=2rx+12πr2
=r[10−(π+2)r]+12πr2
=10r−(12π+2)r2
For maximum area dAdr=0 and d2Adr2 is -ve.
⇒10−(π+4)r=0
(π+4)r=10
r=10π+4
Step 3:
d2Adr2=−(π+4)[Differentiating with respect to r]
(i.e)d2Adr2 is -ve for r=10π+4
⇒A is maximum.
From (1) we have
⇒10=(π+2)r+2x
Put the value of r in (1)
10=(π+2)×(10π+4)+2x
10=10(π+2)π+4+2x
=10(π+2)+2x(π+4)π+4
10(π+4)=10(π+2)+2x(π+4)
10(π+4)−10(π+2)=2x(π+4)
10π+40−10π−20=2x(π+4)
20=2x(π+4)
10=x(π+4)
x=10π+4
Step 4:
Length of rectangle=2r=2(10π+4)
=20π+4
breadth=10π+4