Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to soive the following problem. The volume of a cylinder is 36 cm^3 What is the volume of a sphere if its radius is the same as the cylinder's and the height of the cylinder is equal to the sphere's diameter?
(1 point)
The relationship among the formulas for the volumes of cones, cylinders, and spheres is that the volume of a cone is one-third the volume of a cylinder with the same radius and height, and the volume of a sphere is four-thirds the volume of a cylinder with the same radius.
Given that the volume of the cylinder is 36 cm^3, let's denote the radius of the cylinder as r and the height of the cylinder as 2r (since the height of the cylinder is equal to the sphere's diameter, and the diameter is twice the radius).
The volume of the cylinder is given by the formula: V_cylinder = π * r^2 * h_cylinder
Since the volume of the cylinder is 36 cm^3, we can write: 36 = π * r^2 * 2r
Simplifying the equation, we have: 36 = 2π * r^3
Dividing both sides of the equation by 2π, we get: 18/π = r^3
Now, let's find the volume of the sphere with the same radius as the cylinder. The volume of the sphere is given by the formula: V_sphere = (4/3) * π * r^3
Substituting the value of r^3 from the earlier equation, we have: V_sphere = (4/3) * π * (18/π)
Simplifying further, we get: V_sphere = 4 * 6
Therefore, the volume of the sphere is 24 cm^3.
The volume of a cone is 25/3 cm^3. What is the volume of a sphere if its radius is the same as the cone's and the height of the cone is equal to the
sphere's diameter? (1 point)
The relationship among the formulas for the volumes of cones, cylinders, and spheres is that the volume of a cone is one-third the volume of a cylinder with the same radius and height, and the volume of a sphere is four-thirds the volume of a cylinder with the same radius.
Given that the volume of the cone is 25/3 cm^3, let's denote the radius of the cone as r and the height of the cone as 2r (since the height of the cone is equal to the sphere's diameter, and the diameter is twice the radius).
The volume of the cone is given by the formula: V_cone = (1/3) * π * r^2 * h_cone
Since the volume of the cone is 25/3 cm^3, we can write: 25/3 = (1/3) * π * r^2 * 2r
Simplifying the equation, we have: 25/3 = (2/3) * π * r^3
Dividing both sides of the equation by (2/3) * π, we get: 25 / ((2/3) * π) = r^3
Now, let's find the volume of the sphere with the same radius as the cone. The volume of the sphere is given by the formula: V_sphere = (4/3) * π * r^3
Substituting the value of r^3 from the earlier equation, we have: V_sphere = (4 / 3) * π * (25 / ((2/3) * π))
Simplifying further, we get: V_sphere = (4/3) * ((25 * 3) / (2 * π))
V_sphere = (4/3) * (75 / (2 * π))
V_sphere = (4 * 75) / (3 * 2 * π)
V_sphere = 100 / π
Therefore, the volume of the sphere is 100 / π cm^3.
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)
The volume of the cylinder before removing the cone is 54 in^3.
Let's denote the radius of the cylinder and cone as r and the height as h.
The volume of the cylinder is given by the formula: V_cylinder = π * r^2 * h
Since the volume of the cylinder is 54 in^3, we can write: 54 = π * r^2 * h
Now, let's find the volume of the cone. The volume of the cone is given by the formula: V_cone = (1/3) * π * r^2 * h
Since the cone and cylinder share the same radius and height, the volume of the cone is also 54 in^3.
To find the volume of the remaining part, we subtract the volume of the cone from the volume of the cylinder:
Volume of remaining part = Volume of cylinder - Volume of cone
Volume of remaining part = 54 - 54
Volume of remaining part = 0 in^3
Therefore, the volume of the amount remaining is 0 in^3.
O 40 in.^3
O 27 In.^3
O 18 in.^3
O 36 in.^3
The correct answer is O. 36 in.^3.
The volume of the cylinder before removing the cone is 54 in.^3. The volume of the cone is also 54 in.^3. When the cone is carved out of the cylinder, the remaining volume is equal to the volume of the cylinder minus the volume of the cone.
Therefore, the volume of the remaining part is 54 in.^3 - 54 in.^3 = 0 in.^3.
None of the answer options provided match the correct answer.
Just choose the right one
I apologize for the confusion. The correct answer is O. 36 in.^3.
The volume of the remaining part is equal to the volume of the cylinder minus the volume of the cone.
Given that the volume of the cylinder before removing the cone is 54 in.^3 and the volume of the cone is also 54 in.^3, we have:
Volume of remaining part = Volume of cylinder - Volume of cone
Volume of remaining part = 54 in.^3 - 54 in.^3
Volume of remaining part = 0 in.^3
Therefore, the volume of the amount remaining is 0 in.^3.