Given:
A triangle with vertices A(0,0), B(6,0), and C(0,9) is rotated 360 about the y-axis.
To find:
The volume of the figure.
Solution:
Points A(0,0) and C(0,9) lies on the y-axis. Length of AC is 9 units.
Points A(0,0) and B(6,0) lies on the x-axis. Length of AB is 6 units.
If a triangle with vertices A(0,0), B(6,0), and C(0,9) is rotated 360 about the y-axis, then it will form a cone with radius 6 units and height 9 units.
Volume of a cone is
![V=\dfrac{1}{3}\pi r^2h](https://tex.z-dn.net/?f=V%3D%5Cdfrac%7B1%7D%7B3%7D%5Cpi%20r%5E2h)
where, r is radius of the base and h is vertical height of the cone.
Putting r=6 and h=9, we get
![V=\dfrac{1}{3}\pi (6)^2(9)](https://tex.z-dn.net/?f=V%3D%5Cdfrac%7B1%7D%7B3%7D%5Cpi%20%286%29%5E2%289%29)
![V=\pi (36)(3)](https://tex.z-dn.net/?f=V%3D%5Cpi%20%2836%29%283%29)
![V=108\pi](https://tex.z-dn.net/?f=V%3D108%5Cpi)
Therefore, the volume of the figure is
sq. units.