Arectangle is constructed with its base on the diameter of a semicircle with radius 23 and with its two other vertices on the semicircle. what are the dimensions of the rectangle with maximum area?

     

The area of the rectangle is 529 unit square.        

Step-by-step explanation:

Given : A rectangle is constructed with its base on the diameter of a semicircle with radius 23 and with its two other vertices on the semicircle.

To find : What are the dimensions of the rectangle with maximum area?

Solution :

Let origin be the center of the circle with radius 23.

The equation of the semicircle is

Let (z,0) and (-z,0) be the points on the diameter of the semicircle.  

∴

Since the semicircle in on y-axis so coordinates of point on the semicircle are

So, The length of the rectangle is

Breadth of the rectangle is  

Area of the rectangle is

To maximum we have to derivate w.r.t z to find critical point.

Substitute the value of z in the area and dimensions.

Length is

Breadth is

Area is

Therefore, The area of the rectangle is 529 unit square.

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hope this helped :)