Answer to Question #271512 in Calculus for Emma

Question #271512

The drawing below shows a square with side a. A straight line intersects the square and encloses

an area A. The heights x and y on the left and right side (in a distance d from the square) of

the intersecting line can be varied. Assuming that x  y and x; y  a, nd an expression for

the enclosed area A(x; y) with respect to x and y.



1
Expert's answer
2021-12-15T05:11:19-0500


Area of the trapezoid BCDEBCDE



A=BE+CD2EDA=\dfrac{BE+CD}{2}\cdot EDBMFM=CNFN=GKFK\dfrac{BM}{FM}=\dfrac{CN}{FN}=\dfrac{GK}{FK}

Substitute



BExd=CDxa+d=yxa+2d\dfrac{BE-x}{d}=\dfrac{CD-x}{a+d}=\dfrac{y-x}{a+2d}d(BEx)+a(BEx)=d(CDx)d(BE-x)+a(BE-x)=d(CD-x)d=a(BEx)CDBEd=\dfrac{a(BE-x)}{CD-BE}a+d=a(CDBE+BEx)CDBE=a(CDx)CDBEa+d=\dfrac{a(CD-BE+BE-x)}{CD-BE}=\dfrac{a(CD-x)}{CD-BE}a+2d=a(CDBE+2BE2x)CDBEa+2d=\dfrac{a(CD-BE+2BE-2x)}{CD-BE}=a(CD+BE2x)CDBE=\dfrac{a(CD+BE-2x)}{CD-BE}BExd=CDBEa\dfrac{BE-x}{d}=\dfrac{CD-BE}{a}yxa+2d=(yx)(CDBE)a(CD+BE2x)\dfrac{y-x}{a+2d}=\dfrac{(y-x)(CD-BE)}{a(CD+BE-2x)}

Then



CDBEa=(yx)(CDBE)a(CD+BE2x)\dfrac{CD-BE}{a}=\dfrac{(y-x)(CD-BE)}{a(CD+BE-2x)}CD+BE2x=yxCD+BE-2x=y-xCD+BE2x=y+xCD+BE-2x=y+x

Area of the trapezoid BCDEBCDE



A(x,y)=x+y2a  (units2)A(x, y)=\dfrac{x+y}{2}\cdot a \ \ ({units}^2)

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