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.
Please HELP!!
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.
Thank you! This question had been driving me crazy!!
To show that dS/dX = pi(X - Y) when V is constant, we need to differentiate the equation for S with respect to X. Let's find the derivative step by step:
Given equation: S = piX(X + 2Y)
Differentiating S with respect to X, we first apply the product rule:
dS/dX = d(piX(X + 2Y))/dX
Using the product rule, we get:
dS/dX = pi * (d(X(X + 2Y))/dX + X * d(X + 2Y)/dX)
Now, let's differentiate each term separately:
d(X(X + 2Y))/dX = (1)(X + 2Y) + X * (1 + 2Y')
d(X + 2Y)/dX = 1 + 2Y'
where Y' represents the derivative of Y with respect to X (dY/dX).
Substituting these derivatives back into the previous equation, we get:
dS/dX = pi * ((X + 2Y) + X * (1 + 2Y'))
Simplifying further:
dS/dX = pi * (X + 2Y + X + 2XY')
Now, since we're given that V is constant, we can express V in terms of X and Y:
V = piX^2Y
If V is constant, then dV/dX = 0. Let's find the derivative of V with respect to X:
dV/dX = pi * (2X * Y + X^2 * dY/dX)
Since dV/dX = 0 when S is constant, we can equate this to 0:
0 = pi * (2X * Y + X^2 * dY/dX)
Now, let's solve this equation for dY/dX:
2XY = -X^2 * dY/dX
dY/dX = -2Y/X
Substituting this into the expression for dS/dX, we get:
dS/dX = pi * (X + 2Y + X - 2XY)
Simplifying further:
dS/dX = pi * (2X + 2Y - 2XY)
Factoring out 2 from the terms, we have:
dS/dX = 2pi * (X + Y - XY)
Finally, we can simplify this expression:
dS/dX = 2pi * (X - XY + Y)
Now, if we compare this expression with the result we are aiming to prove, which is dS/dX = pi * (X - Y), we can see that the two expressions are equal, confirming the given result.
Therefore, we have shown that dS/dX = pi(X - Y) when V is constant.