[Math] Rational function to find minimum surface area of cylinder

geometryrational-functions

I have developed the formula to determine the radius of a cylinder with a fixed volume:

$$ f(x) = \sqrt[3]{\dfrac{V}{\pi}}\ $$

Substituted into the formula for the surface area of a cylinder, I get the following function. This would give me the minimum surface area of a cylinder for a given volume.

$$
S(V) = 2\pi(\sqrt[3]{\dfrac{V}{\pi}})^2+2\pi(2 * \sqrt[3]{\dfrac{V}{\pi}})
$$

However, my assignment for class asks for a rational function for this problem. How could I take my existing function and make it rational?

Best Answer

I assume the cylinder in question has $h=r$, so that: $$r=\sqrt[3]{V/\pi}$$ The surface area is then: $$A=2\pi r^2 + 2\pi r h=4\pi r^2=4(\pi V)^{2/3}$$ This of course is not a rational function in $V$ (and never will be), but is a rational function in $r$. Perhaps this is what the assignment means?

Related Solutions

[Math] Why does the cylinder with minimum surface area have a height equal to its diameter

Volume $V=\pi r^2 h$.

Surface area $$S =2 \pi r h +2 \pi r^2$$

Now, the AM-GM inequality says

$$\frac{ \pi r h + \pi rh +2 \pi r^2}{3} \geq \sqrt[3]{\pi r h \cdot \pi rh \cdot2 \pi r^2}= \sqrt[3]{2 \pi V^2 }$$

with equality if and only if $ \pi r h = \pi rh =2 \pi r^2$.

Basically, it boils down to this simple principle (which is more or less AM-GM): if some expressions have a constant product, their sums is minimal if the functions are equal (if they can be equal).

[Math] How to derive the volume and surface area formula for truncated cylinder without calculus

Let $C(A, B)$ denote a truncated cylinder whose left edge has length $A$ and whose right edge has length $B$. (I'm assuming that it's placed so that its longest and shortest edges are to the left and right).

The volume of $C(A, 0)$ is $\frac{1}{2} \pi r^2 A$. Reason: Stack up $C(A, 0)$ atop $C(0, A)$ to get a cylinder of height $A$. The volume of $C(A, 0)$ is clearly half the volume of this cylinder.
Now for a cylinder C(A, B), where $A > B$, consider the stack

C(0, Q)
C(A, B)
C(B, A)
C(Q, 0)

where $Q = A - B$. THat's a cylinder of height $A + B + (A-B) = 2A$. If we subtract from it the volumes of the top and bottom pieces (which we computed in part 1), we get $$ \pi r^2 (2A) - \pi r^2 (Q) = \pi r^2 (2A - (A-B)) = \pi r^2 (A + B) $$ But the volume we've just computed is twice the volume of the sliced cylinder $C(A,B)$ that we're interested in. So the volume of $C(A,B)$ isjust $$ \pi r^2 \frac{A + B}{2} $$

A rather similar proof works for surface area.

Note that the proof given above works for a cylinder cut by ANY two planes, as long as the longest and shortest "line" of the truncated cylinder are diametrically opposite each other.

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Related Solutions

[Math] Why does the cylinder with minimum surface area have a height equal to its diameter

[Math] How to derive the volume and surface area formula for truncated cylinder without calculus

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