Give a composite shape of quarter of circle and rectangle with a total area of #570 " square feet"# and and the diagonal angle of the rectangle equal to #18.43^0#, calculate the radius?

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1 Answer
Sep 1, 2016

#"Radius"~~22.92 "ft".#

Explanation:

Let #r# be the radius. Then, using the Figure given,

#IH=rcos theta, IA=rsin theta, "where, "theta=18.43^@#.

#:." Area of the Rectangle "IHBA=IH*IA=r^2sin thetacos theta#.

#=r^2/2*sin 2theta=r^2/2*(2tan theta)/(1+tan^2 theta)#

#=(r^2tan theta)/(1+tan^2 theta)#

For #theta=18.43^@, tan theta=1/3#

#"Hence, Area of the Rectangle"=(r^2/3)/(1+1/9)=3r^2/10#

#"Also, Area of the Quarter of the Circle"=1/4*pir^2#

#:. "Total Area of the Composite Shape"=3r^2/10+pir^2/4=r^2/20(6+5pi)#, which is, #570" sq.ft."#

#:. r^2/20(6+5pi)=570#

#:. r^2=(570*20)/(6+5pi)=11400/(6+15.7)=11400/21.7~~525.3456#

#:. r~~22.92 "ft."#