Quote Originally Posted by martindcx1e
so x^2 + 4xh = 1200.
h = (1200 - x^2)/(4x)

Volume = (x^2) * [(1200 - x^2)/(4x)]
Which = 300x - (1/4)x^3

Deriv of that = 300 - (3/4)x^2
x = 20

(20)^2 + (4)(20)(h) = 1200
h = 10
You catch on quick, just make sure you answer the question by stating what exactly the maximum volume is. (Not being nit-picky, it's just an easy thing to forget).

Quote Originally Posted by martindcx1e
Ok so you find the equation for what you want to minimize or maximize. Then you find another equation to let you isolate one variable. Then substitute into the original equation, take the derivative, set it to 0, and solve for that variable. Then plug that answer back into your 2nd equation to find the other variable. Is that pretty much it?
You're very close, but try not to think about it as a linear process of this many steps because every problem is different. The process of what you're doing is really like this:

1. Find the equation for you want to minimize or maximize.
2. Take the derivative and get information that lets you answer the question.

It's just that each step will sometimes be straight-forward and sometimes it won't be. In the box question you had to manipulate an extra variable h, but sometimes you'll be manipulating two or more extra variables, and sometimes you won't be manipulating any variables at all (think of the first example I gave of y = 3 - 4x - x^2 or whatever it was).

The key is that before you can take the derivative of the equation you find, it has to be all of the same variable. Like when we had volume = x * x * h, we can't differentiate that equation for volume yet because it's not all of the same variable. If h was a constant we could do it and it would be V' = 2hx but h is a function of x so we can't.

So instead of thinking of it as a linear process, maybe think of it as a series of goals or objectives. How you achieve those objectives might be different going from problem to problem; note the differences in the fence problem from the box program for an example of this. So, onward:

2) Find the point on the line y = 4x + 7 that is closest to the origin.
You're minimizing distance, so your first objective is to get an equation for distance in terms of one variable.