Ratio of variables for expression to have maximum value

Question

I came across this problem in mensuration: A right circular cylinder is to be made out of a metal sheet such that the sum of its height and radius does not exceed 9cm. The cylinder can have a maximum volume of: (a) 54π $cm^3$ (b) 108π $cm^3$ (c) 81π $cm^3$ (d) None of these

The solution given was the following: "To solve this question, one must know that for $a^2*b^3*c^4$ to have the maximum value when (a + b + c) is constant, a,b and c must be in the ratio 1:2:3. Volume of a cylinder = π$r^2$*h. For volume to be maximum, $r^2$h needs to be maximum under the condition that r + h = 9. By the information given above, this is only possible when r : h = 2 : 1, that is, r = 6, h = 3. so, volume of the cylinder = π * 6 * 6 * 3 = 108π". I understand why r = 6, and h = 3. With r + h = 9, the possible value pairs of (r, h) would be {(1, 8), (2, 7), (3, 6), (4, 5), (5, 4), (6, 3), (7, 2), (8, 1)}. Therefore, the respective $r^2$h values would be { 8, 28, 54, 80, 100, 108, 98, 64}. The largest possible volume of the cylinder would be 108π $cm^3$.

1. I want to know if there is a way to find the ratio r : h without enumerating the list of possible value pairs that (r, h) can take.
2. Similar to the above question, I want to know why a : b : c would be 1:2:3 for $a^2b^3c^4$ to be maximum. My understanding so far has been: Let a + b + c = k. Then, $\frac {a}{2} + \frac {a}{2} + \frac {b}{3} + \frac {b}{3} + \frac {b}{3} + \frac {c}{4} + \frac {c}{4} +\frac {c}{4} + \frac {c}{4} = k$ By the AM-GM inequality, we can say that $\frac{1}{9} ( \frac {a}{2} + \frac {a}{2} + \frac {b}{3} + \frac {b}{3} + \frac {b}{3} + \frac {c}{4} + \frac {c}{4} +\frac {c}{4} + \frac {c}{4}) >= ( \frac {a}{2} * \frac {a}{2} * \frac {b}{3} * \frac {b}{3} * \frac {b}{3} * \frac {c}{4} * \frac {c}{4} * \frac {c}{4} * \frac {c}{4})^\frac{1}{9}$ This implies, $\frac{k}{9} >= (\frac{a^2b^3c^4}{2^2*3^3*4^4})^\frac{1}{9}$ This implies, $\frac{k^9}{9^9} >= \frac{a^2b^3c^4}{2^2*3^3*4^4}$. Now,the maximum value that $a^2b^3c^4$ takes when a + b + c is constant is given by: $\frac{4^5}{3 * 9 ^7} * k^9 >= a^2b^3c^4$. How can I get the ratio of a:b:c from here?

Answer 1

This sort of constrained optimization problem can be solved using Lagrange multipliers. In the present case, this is easier if we maximize the logarithm of the volume, since that decouples the variables.

Then the objective function is $2\log r+\log h+\lambda(r+h)$, and setting the derivatives with respect to $r$ and $h$ to $0$ yields

\begin{eqnarray} \frac2r+\lambda&=&0\;,\\ \frac1h+\lambda&=&0\;. \end{eqnarray}

Eliminating $\lambda$ yields $2h=r$ as expected.

It seems you have a typo in your other example – unless I’m mistaken, applying this approach to $a^2b^3c^4$ shows that $a:b:c$ is $2:3:4$, not $1:2:3$.