Optimizing Area in a Semi-Circle

[James99x]

1. Find the dimensions of the rectangle with the largest area that can be inscribed in the upper semi-circle given by x^2+y^2 ≤ 16, y≥0.
2. I thought I'd use A=lw
3. This is but a guess..so take it with a grain of salt..

height=2x
base= x^2+y^2

A(x) = 2x(x^2+y^2)
= 2x^3+2xy^2

A'(x) = 6x^2+2y^2+4x(dy/dx)(y)

I'm not sure what to really do about this particular problem.

 

[Ray Vickson]

Hint: start by drawing a diagram. You will see right away what you did wrong.

RGV

 

[Winning]

Are you familiar with LaGrange multipliers? If so, then this problem is easier than it seems.

 

[I like Serena]

Hi James99x! Welcome to PF!

You have the wrong "base" for your square...

Beyond that your method is fine!