Optimization of a rectangle's area in two parabolas

[cinematic]

Homework Statement

Determine the maximum area of a rectangle formed in the region formed by the two curves
y1=x² - k
y2=x² + k

Homework Equations

The equations are given, I tried using k=1. so y1= x² - 1, etc.

The Attempt at a Solution

Is it true that the rectangle has to be sideways (i.e. NOT parallel to the axes in order to maximize the area of the rectangle?)

 

[LCKurtz]

Why take k = 1? Just leave it as k. Is this problem from a typical first year calculus text? The reason I ask is I would be very surprised if it is asking you to consider rectangles not parallel to the axes. I don't know offhand if the maximum would be attained by a slanted rectangle since I have never seen that problem worked, worked it myself, or even asked for that matter.

 

[cinematic]

Actually, this is an extension on first-time calculus. Our teacher was suggesting if it was possible that by tilting the rectangle, so to speak, the maximized area would be obtained, or if it was in the question's best interest to assume a rectangle with sides parallel to the x-axis.

 

[LCKurtz]

Actually, this is an extension on first-time calculus. Our teacher was suggesting if it was possible that by tilting the rectangle, so to speak, the maximized area would be obtained, or if it was in the question's best interest to assume a rectangle with sides parallel to the x-axis.

I didn't know a question had a "best interest"

The difficulty of having that problem in a first course is that it would be an optimization problem in two variables. If you haven't had that yet, it would presumably be beyond your abilities. Still, it might be an interesting problem given you had the background.