Optimization

[UWMpanther]

1. The problem statement, all variables and given/known data
Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.


2. Relevant equations

(x-a)^2 + (y-b)^2 = r^2
max area = 2x(2y)
= 4xy
3. The attempt at a solution

(x-a)^2 + (y-b)^2 = r^2
= y=r-(x-a)+b

I then plug this into the max area
= 4x(r-(x-a)+b)

I know I need to differentiate, but I'm not sure how to go about this. I know I need to use the product rule, if its setup properly.

[HallsofIvy]

1. The problem statement, all variables and given/known data
Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.


2. Relevant equations

(x-a)^2 + (y-b)^2 = r^2
max area = 2x(2y)
= 4xy
3. The attempt at a solution

(x-a)^2 + (y-b)^2 = r^2
= y=r-(x-a)+b
How in the world did you get this? Surely not by taking the square root of both sides!

I then plug this into the max area
= 4x(r-(x-a)+b)

I know I need to differentiate, but I'm not sure how to go about this. I know I need to use the product rule, if its setup properly.
No, it not at all set up properly!

[UWMpanther]

Ok wow I can't believe I did that.

So I FOIL it out and get this

x^2 = r^2 + 2ax - a^2 - y^2 + 2by - b^2

x= r-a-b+sqrt(2ax)+sqrt(2by)

Does this look better?