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?(1 point) Responses sphere = cylinder + cone sphere = cylinder + cone cone = cylinder – sphere cone = cylinder – sphere cylinder = sphere – cone cylinder = sphere – cone cone = sphere – cylinder cone = sphere – cylinder

Answer 1

[AI-generated answer — Bot GPT 3.5]

sphere = cylinder + cone

Answer 2

Question 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?(1 point) Responses 18 cm3 18 cm cubed 24 cm3 24 cm cubed 12 cm3 12 cm cubed 54 cm3

Answer 3

[AI-generated answer — Bot GPT 3.5]

sphere = cylinder + cone

Given that the volume of the cylinder is 36 cm^3:
Volume of the cylinder = πr^2h
36 = πr^2h
36 = πr^2(2r) [height of the cylinder is equal to the sphere's diameter, which is 2 times the radius]
36 = 2πr^3
r^3 = 18/π
r = (18/π)^(1/3)

Now, we need to find the volume of the sphere with the same radius (r):
Volume of a sphere = (4/3)πr^3
Volume = (4/3)π[(18/π)^(1/3)]^3
Volume = (4/3)π((18/π)^(1/3))^3
Volume = (4/3)π(18/π)
Volume = (4/3)(18)
Volume = 24 cm^3

Therefore, the volume of the sphere is 24 cm^3.

Answer 4

The volume of a cone is 253 π cm3. 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) Responses 252 π cm3 Start Fraction 25 over 2 End Fraction pi cm cubed 256 π cm3 Start Fraction 25 over 6 End Fraction pi cm cubed 25 π cm3 25 pi cm cubed 503 π cm3

Answer 5

[AI-generated answer — Bot GPT 3.5]

Given that the volume of the cone is 253π cm^3:

Volume of the cone = (1/3)πr^2h
253π = (1/3)πr^2h
253π = (1/3)πr^2(2r) [height of the cone is equal to the sphere's diameter, which is 2 times the radius]
253π = (2/3)πr^3
r^3 = 759/π
r = (759/π)^(1/3)

Now, we need to find the volume of the sphere with the same radius (r):
Volume of a sphere = (4/3)πr^3
Volume = (4/3)π[(759/π)^(1/3)]^3
Volume = (4/3)π((759/π)^(1/3))^3
Volume = (4/3)π(759/π)
Volume = (4/3)(759)
Volume = 1012π cm^3

Therefore, the volume of the sphere is 1012π cm^3.

Answer 6

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 36 in.3 36 in. cubed 27 in.3 27 in. cubed 40 in.3 40 in. cubed 18 in.3

Answer 7

[AI-generated answer — Bot GPT 3.5]

Given that the volume of the cylinder before removing the cone is 54 in^3:

Volume of the cylinder = πr^2h
54 = πr^2h
54 = πr^2(2r) [height of the cylinder is equal to the radius since it is a cone carved out of a cylinder]
54 = 2πr^3
r^3 = 27/π
r = (27/π)^(1/3)

Now, the volume of the remaining part after carving out the cone is:
Volume of the remaining = 54 - (1/3)πr^2h
Volume of the remaining = 54 - (1/3)π(27/π)^(2)*27/π
Volume of the remaining = 54 - (1/3)π*729/π^3
Volume of the remaining = 54 - 243/π^2
Volume of the remaining = 54 - 243/π^2
Volume of the remaining = 54 - 243/(3.14159)^2
Volume of the remaining ≈ 36 in^3

Therefore, the volume of the amount remaining after carving out the cone is approximately 36 in^3.

Answer 8

An engineer wants to know the volume of the following object composed of a hemisphere and a cone. She knows the height of the cone is equal to the diameter of the hemisphere, and the volume of the cone is 12 m3 . What is the volume of the whole object?(1 point) Responses 24 m3 24 m cubed 36 m3 36 m cubed 18 m3 18 m cubed There is not enough information to say.

Answer 9

[AI-generated answer — Bot GPT 3.5]

We know the volume of the cone is 12 m^3, and the height of the cone is equal to the diameter of the hemisphere.

Volume of a cone = (1/3)πr^2h
Given that the volume of the cone is 12 m^3, we have:
12 = (1/3)πr^2h
12 = (1/3)πr^2(2r) [height of the cone is equal to the diameter]
12 = (2/3)πr^3
r^3 = 18/π
r = (18/π)^(1/3)

Now, we know that the height of the cone is equal to the diameter of the hemisphere. Hence, the radius of the hemisphere is also (18/π)^(1/3).

The volume of the hemisphere is (2/3)πr^3.
Volume of the hemisphere = (2/3)π[(18/π)^(1/3)]^3
Volume of the hemisphere = (2/3)π(18/π)
Volume of the hemisphere = 12 m^3.

Therefore, the total volume of the object (hemisphere + cone) = Volume of the hemisphere + Volume of the cone = 12 m^3 + 12 m^3 = 24 m^3.

The volume of the whole object is 24 m^3.