It's been a while since I've taken calculus, so I don't remember how to optimize three-variable problems. Can anyone help?

Maximize (x*y*z) given:

100 ≥ x ≥ y ≥ z ≥ 0

200 ≥ x + pi*sqrt(y^2 + z^2)

(In case anyone's wondering, I'm trying to figure out what shape of box I should use to maximize the amount of stuff I can ship home in one box given the size restrictions for what Austria post will ship internationally.)

## A math problem (maximization with restraints)

**Moderators:** gmalivuk, Moderators General, Prelates

- SirGabriel
**Posts:**42**Joined:**Wed Jul 16, 2014 11:54 pm UTC

### A math problem (maximization with restraints)

Last edited by gmalivuk on Thu Mar 31, 2016 12:24 pm UTC, edited 1 time in total.

**Reason:***added a description to the topic title*### Re: A math problem

If you take x as fixed you could optimize it for two variables first as a function of x:

max(y*z) such that [(200-x)/pi]

I think this has a nice symmetric solution.

Then you maximize it separately for x alone.

max(y*z) such that [(200-x)/pi]

^{2}≥(y^2 + z^2)I think this has a nice symmetric solution.

Then you maximize it separately for x alone.

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- ThirdParty
**Posts:**351**Joined:**Wed Sep 19, 2012 3:53 pm UTC**Location:**USA

### Re: A math problem (maximization with restraints)

I agree. x = 200/3, y = z = (200/3)(√2/π).

Here's my reasoning:

Here's my reasoning:

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