Optimization Question! Please help calculus 1

Question

Find the dimensions of the rectangle of maximum area that can be inscribed in a circle of radius r=21.

 

The rectangle of maximum area has dimensions:?

 

 

The picture is of a rectangle with a line going half way through it from the top right hand corner saying R. I know you have to double the line cutting the rectangle into two triangles.  Please help. Hw Is due soon. 

 

Answer 1

OK. The area of the rectangle - the thing you want to maximize - is LW. Length times width. A = LW.

 

But since the thing is inscribed in a circle, the rectangle's diagonal must equal the circle's diameter.

This is described by L² + W² = (2R)² = 4R² = 1764 in².

 

We can solve this constraint for either L or W. I solved it for W.

 

W² = 1764 - L² or W = (1764 - L²)^(1/2)

 

And substitute this for W in the equation for area:

 

A = L (1764 - L²)^(1/2)

 

A is maximized when dA/dL = 0.

 

dA/dL = (1764 - L²)^(1/2) - (1764 - L²)^(-1/2)L²

= (1764 - L²)^(-1/2) [(1764 - L²) - L²]

 

= 0 if 1764 - 2L² = 0

 

Heck with solving this for L by hand. I used Wolfram Alpha. And got

 

L = 29.7

 

W = 29.7 = L

 

The inscribed rectangle with the maximum area is a square. Cheers!