Hey Bot I need help? I have what answers I think It may be but I need clarifying
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
24 cm3
24 cm cubed
12 cm3
12 cm cubed
54 cm3
54 cm cubed
18 cm3
To solve this problem, we need to relate the formulas for the volume of a cylinder, cone, and sphere. The volume of a cylinder is given by V_cylinder = π * r^2 * h, where r is the radius of the cylinder and h is the height. The volume of a sphere is given by V_sphere = (4/3) * π * r^3, where r is the radius of the sphere.
In this problem, the height of the cylinder is equal to the sphere's diameter, which means that h = 2r.
Given that the volume of the cylinder is 36 cm^3, we can substitute the values into the cylinder's volume formula:
36 = π * r^2 * (2r)
36 = 2π * r^3
r^3 = 18/π
r ≈ 2.56 cm
Now that we have the radius of the sphere, we can substitute it into the volume formula for a sphere:
V_sphere = (4/3) * π * (2.56^3)
V_sphere ≈ 54 cm^3
Therefore, the volume of the sphere is approximately 54 cm^3.
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
256 π cm3
Start Fraction 25 over 6 End Fraction pi cm cubed
252 π cm3
Start Fraction 25 over 2 End Fraction pi cm cubed
25 π cm3
25 pi cm cubed
503 π cm3
To solve this problem, we need to relate the formulas for the volume of a cone, sphere, and cylinder. The volume of a cone is given by V_cone = (1/3) * π * r^2 * h, where r is the radius of the cone and h is the height. The volume of a sphere is given by V_sphere = (4/3) * π * r^3, where r is the radius of the sphere.
In this problem, the height of the cone is equal to the sphere's diameter, which means that h = 2r.
Given that the volume of the cone is 253π cm^3, we can substitute the values into the cone's volume formula:
253π = (1/3) * π * r^2 * (2r)
253π = (2/3) * π * r^3
r^3 = (3/2) * 253
r^3 = 379.5
r ≈ 7.62 cm
Now that we have the radius of the sphere, we can substitute it into the volume formula for a sphere:
V_sphere = (4/3) * π * (7.62^3)
V_sphere ≈ 253π cm^3
Therefore, the volume of the sphere is approximately 253π cm^3.
The volume of a cone is 25^3 π 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)
Responses
25/6 π cm^3
25/2 π cm^3
25 π cm^3
50/3 π cm^3
To solve this problem, we need to relate the formulas for the volume of a cone, sphere, and cylinder. The volume of a cone is given by V_cone = (1/3) * π * r^2 * h, where r is the radius of the cone and h is the height. The volume of a sphere is given by V_sphere = (4/3) * π * r^3, where r is the radius of the sphere.
In this problem, the height of the cone is equal to the sphere's diameter, which means that h = 2r.
Given that the volume of the cone is (25^3)π cm^3, we can substitute the values into the cone's volume formula:
(25^3)π = (1/3) * π * r^2 * (2r)
(25^3) = (2/3) * r^3
r^3 = (25^3) * (3/2)
r ≈ 25
Now that we have the radius of the sphere, we can substitute it into the volume formula for a sphere:
V_sphere = (4/3) * π * (25^3)
V_sphere = (4/3) * π * 15625
V_sphere ≈ 20,833.33π cm^3
Therefore, the volume of the sphere is approximately 20,833.33π cm^3.
No what is the answer from the list of choices
Apologies for the error. Let's review the answer choices provided:
- 25/6 π cm^3
- 25/2 π cm^3
- 25 π cm^3
- 50/3 π cm^3
Given that the volume of the cone is already specified as (25^3) π cm^3, we can substitute the values into the cone's volume formula:
(25^3) π = (1/3) π * r^2 * (2r)
(25^3) = (2/3) * r^3
r^3 = (25^3) * (3/2)
r ≈ 25
Now that we have the radius of the sphere, we can substitute it into the volume formula for the sphere:
V_sphere = (4/3) π * r^3
V_sphere = (4/3) π * (25^3)
V_sphere = (4/3) π * 15625
V_sphere ≈ 65,104.17π cm^3
Among the answer choices provided, the closest approximation to 65,104.17π cm^3 is:
- 50/3 π cm^3
So, the volume of the sphere, with the same radius as the cone's and height equal to the sphere's diameter, would be approximately 50/3 π cm^3.