3. Suppose you need to cut a beam with maximal rectangular cross section from an

Question

3. Suppose you need to cut a beam with maximal rectangular cross section from an elliptical log with semi-axes of lengths 2 ft and 1 foot. Use Lagrange multipliers to find the maximal cross-sectional area of such a beam. Include correct units in your answer. (If coordinate axes are set up so the center of the log is at the origin, the log is thenbounded by the ellipse x2+4y2=4 See the figure below).

I sort of know what to do, but I don't know where to go from what I have written. any help would be great thanks!

Explanation / Answer

x2+4y2=4 is equation of ellipse

g(x,y)=x2+4y2-4

area of crosssection =(x*(2y))=2xy

f(x,y)=2xy

f=<2y,2x>

g=<2x,8y>

f=g

<2y,2x>=<2x,8y>

2y=2x ,2x=8y

==>=y/x ,=x/4y

y/x =x/4y

==>x2-4y2=0

we already have x2+4y2=4

solving x2+4y2=4,x2-4y2=0

x2+4y2+x2-4y2=4+0

2x2=4

x=2

x2-4y2=0

(2)2-4y2=0

y2=1/2

y=1/2

weare not taking negative values of x, y because area is positive

maximum cross sectional area A=2xy

A=2*2 *1/2

A=2

maximum cross sectional area A=2 ft2

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