Thread: 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

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.



So from here just take the derivative.

Also the the identity

So