Problem 10

A paper drinking cup filled with water has the shape of a cone with height and semi-vertical angle .

A ball is placed carefully in the cup, thereby displacing some of the water and making it overflow.

Step 1.

We position the cone horizontally, with its tip at the origin, as shown in the diagram.

Let, the radius of the ball be and the coordinate of its centre be .

We have the volume of displaced water, which is the volume of revolution obtained by rotating the upper section of the circle.

           

The equation of the circle is and the upper half is the graph of the function  

Using the washer method, we find the volume of displaced water