How to solve such an optimization problem efficiently？?

How to solve optimization problem using only pre-calc (no calc and no calculator)?

• A manufacturer wants to make wooden crates with square base, no top and capacity of 32 ft^3. Find the dimensions that minimize the amount of wood used. I can do this problem with a calculator. But without one, I'm totally stuck. Really all i need help in is minimizing the equation y = x^2 + 128/x

• Answer:

  Dear Person, I agree with your work, that you really just need to minimize the function: f(x) = x^2 + 128/x where x is interpreted as the side length of the square base, the height h satisfies hx^2 = 32, and f(x) is the total area of the wood. It is not at all obvious how to minimize f(x) without calculus. However, from calculus, you can easily get that the minimum is at x=4 (and so height is 2). A non-calculus answer can be done, but is more complex. Let's use the following fact: Fact: If x>0, then x + 1/x >= 2. Proof: Since x>0, x + 1/x >=2 is equivalent to x^2 + 1 >= 2x, which is equivalent to x^2 - 2x + 1 >= 0, which is equivalent to (x-1)^2 >= 0, which is trivially true since a perfect square is always non-negative. Corollary: If x > 0, then 4/x + x/4 >= 2 Proof: Just define z = x/4, and use the previous fact. Now for your function f(x): For any value x>0, we have: f(x) = x^2 + 128/x = x^2 + 32(4/x) = x^2 + 32(4/x + x/4 - x/4) = x^2 + 32(4/x + x/4) - 8x = x^2 - 8x + 32(4/x + x/4) >= x^2 - 8x + 32*2 [where this part uses the corollary above] = x^2 - 8x + 16 -16 + 64 = (x-4)^2 + 64 -16 >= 64 - 16 [where this part uses the fact that (x-4)^2 >= 0 ] = 48. And this minimum value 48 is achieved at x=4. ***Note: In this derivation, I "cheated" in that I used calculus to get the answer x=4. Then, knowing the answer, I constructed the above non-calculus proof.