# calc

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given S=piX(X+2Y) and V=piX^2Y show that dS/dX=pi(X-Y) when V is constant, and
dV/dX=-piX(X-Y)when S is constant.

I have tried several times, and I don't get the result you asked for.

If we're given S=piX(X+2Y), then
dS/dx= piX(1+2Y')+pi(X+2Y)=
pi(X+2XY'+X+2Y) or
(1) dS/dx = 2pi(X+Y+XY')
We're also told V=piX^2Y so
dV/dx=piX^2Y' + 2piXY or
(2) dV/dx = piX(XY'+2Y)
(You should recognize these as the surface and volume formulas for a right circular cylinder.)
When V is constant (2) is 0, so XY'+2Y=0 or Y'=-2Y/X. Using this for Y' in (1) we get
dS/dx = 2pi(X+Y+X(-2Y/X))=2pi(X-Y)
Are you missing a 2 or did I accidentally add one, check to see if this is correct. Also note that the result holds when X>0, for X=0 the result is trivial.

You should be able to do a similar calculation when dS/dx = 0, solve for Y' and substitute into dV/dx.

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