# Thread: Largest rectangle that can inscribed in a semicircle?

1. ## Largest rectangle that can inscribed in a semicircle?

Determine the area of the largest rectangle that can be inscribed in a semicircle of radius 8". Figure shows that the area can be written as A = (2x)y, if (x,y) is the point of the upper right corner of the rectangle. However, we choose to parameterize the area by a single value, the angle "theta".

1)
Derive the formula for the area of the inscribed rectangle as a function of theta. We refer to this function as A(theta) below.

A(theta) = ?

So I guess the area is: (2*x) * (sqrt(16-x^2))?

2)
Plot A(theta) over its relevant domain. What is the relevant domain?

I have no idea what the domain is, is it [0, 8]?

3)For what value of theta is the maximum area attained?

4)What is the maximum area?

I know I have to take the 1st derivative, but I don't know if my area is set up right. any help is greatly appreciated

2. Originally Posted by Lunar
Determine the area of the largest rectangle that can be inscribed in a semicircle of radius 8". Figure shows that the area can be written as A = (2x)y, if (x,y) is the point of the upper right corner of the rectangle. However, we choose to parameterize the area by a single value, the angle "theta".

1) Derive the formula for the area of the inscribed rectangle as a function of theta. We refer to this function as A(theta) below.

A(theta) = ?

So I guess the area is: (2*x) * (sqrt(16-x^2))?

2) Plot A(theta) over its relevant domain. What is the relevant domain?

I have no idea what the domain is, is it [0, 8]?

3)For what value of theta is the maximum area attained?

4)What is the maximum area?

I know I have to take the 1st derivative, but I don't know if my area is set up right. any help is greatly appreciated
See the attached picture

The area of the whole rectangle will be 4 times the area of of the rectangle in the first quadrant.

$\displaystyle A(\theta)=4(8\cos(\theta)(8\sin(\theta)=256\sin(\t heta)\cos(\theta)$

So from here just take the derivative.

Also the the identity $\displaystyle 2\sin(\theta)\cos(\theta)=\sin(2\theta)$

So $\displaystyle A(\theta)=128\sin(2\theta)$