A cap of a sphere is generated by rotating the region about the y-axis. Determine the volume of this cap when the radius of the sphere is 5 inches and the height of the cap is 1 inch.
To find the volume of the cap of a sphere, we need to consider the concept of a spherical cap.
A spherical cap is a portion of a sphere that is cut by a plane parallel to its base. In this case, the base of the cap is a circle, and the height of the cap is the distance between the base of the cap and the plane that cuts the sphere.
To calculate the volume of the cap, we can use the formula:
V = (1/3) * π * h^2 * (3r - h)
Where:
V is the volume of the cap,
π is a mathematical constant approximately equal to 3.14159,
h is the height of the cap, and
r is the radius of the sphere.
Given the radius (r = 5 inches) and the height of the cap (h = 1 inch), we can substitute these values into the formula to find the volume.
V = (1/3) * π * (1)^2 * (3(5) - 1)
Simplifying this equation:
V = (1/3) * π * (1) * (15 - 1)
V = (1/3) * π * (1) * (14)
V = (14/3) * π
Therefore, the volume of the cap of the sphere is approximately (14/3)π cubic inches.