Hello. I've been stuck on a Lagrange Multiplier problem. It's from Mathematical Methods in the Physical Sciences by Mary Boas 3rd edition pg. 222. The question is:
What proportions will maximize the volume of a projectile in the form of a circular cylinder with one conical end and one flat end, if the surface area is given?
Then there is a picture of a cylinder with a cone attached to the end. the circular base has radius r, cylinder has length l, and slope of the cone is marked s.
So I've started doing the problem, but something just doesn't seem right. How am I supposed to get an answer if I don't know what the surface area is? I looked in the back of the book and there are numerical answers for r, l, and s. How am I supposed to get actual number answers and not something just in terms of the surface area?
I really want to get this clarified before I go much further because the algebra is absolutely horrendous.
The volume is V=pi*r^2*l+(1/3)*pi*r^2*(sqrt(s^2-r^2))
and the surface area is SA=pi*r*s+pi*r^2+2*pi*r*l
then to do lagrange multipliers you write F=V+b(SA)
(usually lambda is used instead of b)
The Attempt at a Solution
So then you get use the three partial derivatives and the surface area equation and get a system of 4 equations and solve.
I've solved them out to get values of r, l, and s as functions of lambda (b) and it was a huge mess. Now all that's left is to plug them into the surface area equation and somehow get numerical answers...? I'm really confused.