Question

Suppose a sphere, cylinder, and cone all share the same radius. The cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?

A. cylinder = sphere - cone
B. cone = sphere -cylinder
C. cone = cylinder - sphere
D. sphere = cylinder + cone

Answer 1

[AI-generated answer — Bot GPT 3.5]

The correct answer is C. cone = cylinder - sphere.

Answer 2

Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 cm3 . 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?

A. 12 cm^3
B. 24 cm^3
C. 18 cm^3
D. 54 cm^3

Answer 3

[AI-generated answer — Bot GPT 3.5]

We know that the volume of a cylinder is given by V_cylinder = πr^2h, where r is the radius and h is the height.

In this case, the volume of the cylinder is 36 cm^3, so we have 36 = πr^2h.

We also know that the height of the cylinder is equal to the diameter of the sphere, which means h = 2r.

Substituting this into the equation for the volume of the cylinder, we have 36 = πr^2(2r).

Simplifying this, we get 36 = 2πr^3.

Dividing both sides of the equation by 2π, we have r^3 = 18.

Taking the cube root of both sides, we get r = 3.

Now we can find the volume of the sphere using the formula for the volume of a sphere, V_sphere = (4/3)πr^3.

Substituting r = 3, we have V_sphere = (4/3)π(3^3) = (4/3)π(27) = 36π.

So the volume of the sphere is 36π cm^3. Since π is approximately equal to 3.14, we can estimate the volume of the sphere as 36(3.14) ≈ 113.04.

Therefore, the closest answer choice is D. 54 cm^3.

Answer 4

The volume of a cone is 25/3 pi 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?

A. 50/3 pi cm^3
B. 25 pi cm^3
C. 25/2 pi cm^3
D. 25/6 pi cm^3

Answer 5

[AI-generated answer — Bot GPT 3.5]

We know that the volume of a cone is given by V_cone = (1/3)πr^2h, where r is the radius and h is the height.

In this case, the volume of the cone is (25/3)π cm^3, so we have (25/3)π = (1/3)πr^2h.

We also know that the height of the cone is equal to the diameter of the sphere, which means h = 2r.

Substituting this into the equation for the volume of the cone, we have (25/3)π = (1/3)πr^2(2r).

Simplifying this, we get (25/3)π = (2/3)πr^3.

Dividing both sides of the equation by (2/3)π, we have r^3 = (25/2).

Taking the cube root of both sides, we get r = ∛(25/2) = (25/2)^(1/3).

Now we can find the volume of the sphere using the formula for the volume of a sphere, V_sphere = (4/3)πr^3.

Substituting r = (25/2)^(1/3), we have V_sphere = (4/3)π((25/2)^(1/3))^3.

Simplifying this, we get V_sphere = (4/3)π(25/2) = (100/3)π.

Therefore, the volume of the sphere is (100/3)π cm^3. Since π is approximately equal to 3.14, we can estimate the volume of the sphere as (100/3)(3.14) = 314.16/3.

Simplifying further, we get ≈ 104.72 cm^3.

Therefore, the closest answer choice is A. 50/3 π cm^3.