A figure consists of a semicircle with a rectangle on its diameter . Given the perimeter of the figure find its dimensions in order that the area may be maximum.


let the radius of the circle be r and the height of the rectangular part formed on the diameter of circle be h.
let the area of the figure be 
A=area of semicircle + area of rectangleA=12.πr2+(2r)h .........(1)
since the perimeter of the figure is fixed , let it be 2k.
2k=πr+2r+2hk=πr2+r+hh=k-πr2-r .......(2)
substituting in eq(1):
A=12.πr2+2r*[k-πr2-r]A=12.πr2+2rk-πr2-2r2A=-πr22+2rk-2r2 .......(3)
now to find the critical points let us differentiate eq(3) wrt r:
dAdr=-πr+2k-4r ........(4)put dAdr=0-πr+2k-4r=0(π+4)r=2kr=2kπ+4
now to check  the maxima or minima we will again differentiate the equation (4):
d2Adr2=ddr(dAdr)=ddr(-πr+2k-4r)=-π-4d2Adr2<0
thus r=2kπ+4 must be a relative maximum.
since there is only one critical point , a relative maximum is also an absolute max.
thus r=2kπ+4 gives the maximum area in the problem.
the corresponding value of the height is
h=k-πr2-r=k-π2.2kπ+4-2kπ+4=k-πkπ+4-2kπ+4=kπ+4k-πk-2kπ+4=2kπ+4

hope this helps you
 

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