Question

Find the dimensions of the rectangle of maximum area with sides parallel to the coordinate axes that can be inscribed in the ellipse 4x^2 +16y^2 = 16.
I am totally confused on this one, can someone explain what to do please?

Answer 1

You need to maximize the value of the rectangle, which will be. So you need a formula for that. A = l * w.
If the width is the X value the length will be the y value, so you have:
A = xy, or
A = x * f(x)
So you need to find y in terms of:
y = \[\sqrt{(16-4x ^{2})/16}\]

\[A = x\sqrt{(16-4x ^{2})/16}\]

Then you want to maximize that within the range of 0 < x < 2. This means taking the derivative of A to find a turning point.