Maximizing Area of Rectangle in x+3y=12 Plane

[imsoconfused]

Homework Statement

A rectangle has sides on the x and y axes and a corner on the plane x+3y=12. Find its maximum area.

Homework Equations

A=xy=(12-3y)y

(A=12, according to the solution manual.)

The Attempt at a Solution

At first I thought the corner it was talking about lay on one of the axes, but now I realize that it is a point around (4,3). I know there is a derivative (partial?) I need to take, but I don't know which one and or what to do it with respect to. I've drawn a graph so I can see what I'm doing, but the professor just skimmed over maximixation and I'm really confused! This chapter is not at a point that Lagrange multipliers have been covered. That's the way I would expect to do it, obviously not.

 

[Dick]

Ok, if you want to maximize set dA/dy=0. What's y?

 

[imsoconfused]

dA/dy=12-6y. y=2, and then (12-3(2))*2=12 and that's the area. I knew it couldn't be that hard. thanks!