Volume of Cones, Cylinders, and Spheres Quick Check
4 of 54 of 5 Items
Question
Use the image to answer the question.
A cone is placed inside a cylinder. The apex of the cone touching the center of the top circle of the cylinder is highlighted with a dot. The cone with its base is drawn in dashed lines. The base of the cone is common with the base of the cylinder.
A sculptor created a design by carving a cone out of a cylinder. The cone and cylinder share the same radius and height. If the volume of the cylinder before removing the cone is 54 in.3 , what is the volume of the amount remaining?(1 point)
Responses
18 in.3
18 in. cubed
36 in.3
36 in. cubed
40 in.3
40 in. cubed
27 in.3
27 in. cubed
The volume of the amount remaining after carving out the cone can be found by subtracting the volume of the cone from the volume of the cylinder.
Since the cone and cylinder have the same height and radius, they have the same base area. The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height. The volume of the cylinder is given as 54 in.3, so we can find the radius using the formula.
54 = πr^2h
54 = πr^2(r)
54 = πr^3
r^3 = 54/π
r ≈ 3.24 inches
Now that we have the radius, we can use it to find the volume of the cone. The formula for the volume of a cone is V = (1/3)πr^2h, where r is the radius and h is the height (which is the same as the height of the cylinder).
V_cone = (1/3)π(3.24)^2(3.24)
V_cone ≈ 10.99 in.3
To find the volume of the amount remaining, we subtract the volume of the cone from the volume of the cylinder.
V_remaining = V_cylinder - V_cone
V_remaining = 54 - 10.99
V_remaining ≈ 43.01 in.3
Therefore, the volume of the amount remaining is approximately 43.01 in.3
Answer: 43 in.3