A serving bowl has an inside diameter of 12 inches and a depth of 4 inches. The bowl is fashioned from part of a sphere.
Find the capacity (volume) of the bowl.
just check out the volume of a spherical cap:
http://mathworld.wolfram.com/SphericalCap.html
That website isn't helpful since the problem is a box cross sectional problem. But thank you for trying to help
To find the capacity (volume) of the bowl, we need to calculate the volume of the sphere that the bowl is fashioned from and then subtract the volume of the unfilled portion.
Step 1: Find the volume of the sphere.
The formula for the volume of a sphere is given by:
V = (4/3) * π * r^3
In this case, we know the diameter of the sphere, which is equal to 12 inches. The radius of the sphere (r) is therefore half of the diameter, which is 12/2 = 6 inches.
Plugging in the values, we have:
V_sphere = (4/3) * π * (6)^3
Step 2: Find the volume of the unfilled portion.
The unfilled portion is in the shape of a cone, with the radius (r) equal to half the inside diameter of the bowl (12/2 = 6 inches) and the height (h) equal to the depth of the bowl (4 inches).
The formula for the volume of a cone is given by:
V_cone = (1/3) * π * r^2 * h
Plugging in the values, we have:
V_cone = (1/3) * π * (6)^2 * 4
Step 3: Calculate the volume of the bowl.
The volume of the bowl is obtained by subtracting the volume of the unfilled portion from the volume of the sphere.
V_bowl = V_sphere - V_cone
Substituting the values, we have:
V_bowl = [(4/3) * π * (6)^3] - [(1/3) * π * (6)^2 * 4]
Simplifying the equation further would give you the final answer for the capacity (volume) of the bowl.