Final answer:
To find the volume of the ellipsoid x² + y² + 9z² = 81, we identify the radii along the x, y, and z axes as 9, 9, and 3, respectively, and use the formula V = 4/3 π abc to get a volume of 972 π cubic units.
Step-by-step explanation:
The question involves finding the volume of an ellipsoid with the equation x² + y² + 9z² = 81. An ellipsoid is a 3D shape that is the generalization of a sphere, and its volume can be found using the formula V = 4/3 π abc, where a, b, and c are the radii along the x, y, and z axes, respectively.
For the given ellipsoid, we can see that a = b = 9 and c = 3 because in the equation, the coefficients of x² and y² are 1 (implying a radius of 9 after taking the square root of 81), while the coefficient of 9z² implies a radius of 3 along the z-axis. Therefore, the volume is:
V = 4/3 π (9)(9)(3)
Now, we compute the volume:
V = 4/3 π (729)
V = 4/3 π (243)
V = 972 π cubed units
Thus, the volume of the ellipsoid is 972 π cubic units.