Expert: Scotto - 1/9/2009

Question
QUESTION: I have a problem here which asks me to find the dimensions of the right-circular cylinder of greatest surface area that can be inscribed in a sphere of Radius R (note it is the surface area they are asking about, not the volume).

The answer is even given as
height = 2* (5-(5^1/2)/10)^1/2 R
radius = (55-(5^1/2)/10)^1/2 R

But I have tried geometrically and algebraically for several days, and cannot get anything close to this.

ANSWER: Geometry and algebra don't seem like they can solve it as easily as calculus.

The height of the cyliner is h.
The radius of the cylinder is r.
Take the radius of the sphere as R.

We know that the volume of the cylinder is [1] V = πr²h.

It is also known from drawing a trianle represent a vertical slice of the cylnder through the middle that the [2] r² + (h/2)² = R².

Using formula [2], it can be solved for r².  We now have r² as a function of h.  Since formula [1] contains an r², we don't even need to take a squareroot.  All we have to do is substitute what we got for r² back into equation [1].  We now have V in terms of h alone, so are rewritten equation [1] is really V(h) instead of V(r,h)

If we find V'(h) { a product rule and a power rule } and set it equal to 0, the height h of the cylinder with maximum volume can be found.  Using equation [2] and the value of h, we can also find r.  Using both h and r, we can find V.


---------- FOLLOW-UP ----------

QUESTION: Thanks. But the question is asking about the surface area not the volume,

The surface area of course, is 2 pi r ^2 + 2 pi r h

I used a variation of your formula [2] but got no where (when I said algebraically or geometrically, I meant taking derivatives of algebraic variables or trigonometric variables (like Rsin@)). but will try again.

Regards,

Richard Thieme


Answer
Sorry I missed that.  That is the right formula for the surface area (top, bottom, and side) of a cylinder.

Since the equation is in a sphere, r² + (h/2)² = R² where r is the radius of the cylinder, h is the height of that cylinder, and R is the radius of the circle.  Remembr, R is a constant, so h can be solved for using that equation.

Put h into the equation on the area, take the derivative, and set it to 0.  Once this has been done, solve for r and then you can solve for h.