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
503 π cm3
Start Fraction 50 over 3 End Fraction pi cm cubed
25 π cm3
25 pi cm cubed
256 π cm3
Start Fraction 25 over 6 End Fraction pi cm cubed
252 π cm3
To find the volume of the sphere, we first need to find the radius of the cone. The formula for the volume of a cone is V = (1/3)πr²h, where V is the volume, r is the radius, and h is the height.
Given that the volume of the cone is 253π cm³, we can plug this into the formula:
253π = (1/3)πr²h
Since the height of the cone is equal to the diameter of the sphere, we can substitute h with 2r:
253π = (1/3)πr²(2r)
Simplifying, we have:
253π = (2/3)πr³
Now we can isolate r³ by multiplying both sides by (3/2):
r³ = (3/2)(253π)
r³ = 379.5π
Now we can find the volume of the sphere using the formula V = (4/3)πr³:
V = (4/3)π(379.5π)
V = (4/3)(379.5π²)
V ≈ 505.98π
Therefore, the volume of the sphere is approximately 505.98π cm³.
Therefore, the correct response is 503 π cm³.
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
12 cm3
12 cm cubed
54 cm3
54 cm cubed
18 cm3
18 cm cubed
24 cm3
To find the volume of the sphere, we can use the relationship between the volumes of a cone and a sphere.
The volume of a cylinder is given by V = πr^2h, where V is the volume, r is the radius, and h is the height.
Given that the volume of the cylinder is 36 cm³, we can arrange the formula as follows:
36 = πr^2h
Since the height of the cylinder is equal to the diameter of the sphere, we can substitute h with 2r:
36 = πr^2(2r)
Simplifying, we have:
36 = 2πr^3
Dividing both sides by 2π, we get:
18 = r^3
Now, to find the volume of the sphere, we use the formula V = (4/3)πr^3:
V = (4/3)π(18)
V = 72π/3
V = 24π
Therefore, the volume of the sphere is 24π cm³.
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
To find the volume of the remaining shape after carving out the cone, we first need to find the volume of the cone.
The volume of a cylinder is given by V = πr^2h, where V is the volume, r is the radius, and h is the height.
Given that the volume of the cylinder is 54 in³, we can plug this into the formula:
54 = πr^2h
Since the cone and cylinder share the same radius and height, the volume of the cone will be (1/3) times the volume of the cylinder.
So, the volume of the cone is:
V_cone = (1/3) * V_cylinder
V_cone = (1/3) * 54
V_cone = 18 in³
Now, to find the volume of the remaining shape, we subtract the volume of the cone from the volume of the cylinder:
V_remaining = V_cylinder - V_cone
V_remaining = 54 - 18
V_remaining = 36 in³
Therefore, the volume of the amount remaining is 36 in³.
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
18 m3
18 m cubed
There is not enough information to say.
There is not enough information to say.
36 m3
36 m cubed
24 m3
To find the volume of the entire object composed of a hemisphere and a cone, we need to find the volume of each component separately and then sum them up.
Given that the volume of the cone is 12 m³, we can use the formula for the volume of a cone: V_cone = (1/3)πr²h.
However, the problem does not provide the radius (r) of the cone or the height (h) of the cone, which is equal to the diameter of the hemisphere. Therefore, we cannot calculate the volume of the cone or the entire object.
Hence, the correct response is "There is not enough information to say."