Calculating Total Height of Cylinder with Two Identical Cones on Each End

[bigguns101]

a cylinder has two identical cones one each end. the volume of the object is 50(pie/3)
find the total height.

so 2h+k=height.
where to from there

 

[TOONCES]

It's difficult to picture what you are talking about exactly but I'll take a stab at it.

(pi*r*s+pi*r^2) = surface area of a cone
(2*pi*r*h+2*pi*r^2)= surface area of a cylinder

where s is the length of the cone side, r is the radius, h is the height of the cylinder, and pi is pi.

Since the object is made up of a cone on top of a cylinder on top of a cone, and assuming the cones are pointed outwards from the center of the cylinder, the second term in each of the surface area equations can be ignored. These terms give the surface area of the surfaces that are connected between the three objects. So the new equations are:

pi*r*s=surface area of the cone
2*pi*r*h=surface area of the cylinder

There are two cones so multiply that equation by 2. The sum of the two is the total surface area of the object.

Now there are many techniques to minimize the surface area and I don't know all of them. In order to do so using basic calculus you need to either get one variable in terms of all the others, or apply a constraint on all the variables besides one. So you can get r in terms of h and s or you can say h equals a number and s equals a number and solve that way. That's not it though. After you've done one of the two, you must take the equation for the total surface area of the object and take the derivative. Now set the derivative equal to 0 and solve for the variable left (r in my examples). This r should correspond to either the maximum or minimum value of the surface area of the object.

 

[mathman]

You need to specify two parameters.

V=πr²(m + 2k/3), where r= radius, m=cylinder height, k=cone height.

For fixed V you need to specify two variables among r,k,m.

 

[Grep]

That's a homework style problem if I ever saw one. You'd be best to ask this question in the "Homework and Coursework Questions" section, in the "Calculus and Beyond" sub-forum.

Hint for now: You'll likely need implicit differentiation. Also, I assume the cones are attached on the circular part to the cylinder, so you need not consider the surface area of the ends of the cylinder.

 

[berkeman]

(Thread moved to Homework Help, Calculus & Beyond forum)