Suppose a sphere and a cone share the same radius and the height of the cone is twice the radius. Which ratio describes the relationship between the formulas for the volume of a cone to the volume of a sphere?(1 point)
:
The volume of a cone can be calculated using the formula V_cone = (1/3)πr^2h, where r is the radius and h is the height.
In this case, we are given that the volume of the sphere is 72 m^3. We know that the volume of a sphere is given by the formula V_sphere = (4/3)πr^3.
We are also given that the cone has the same radius as the sphere, so we can write r_cone = r_sphere.
Finally, we are given that the height of the cone is equal to the diameter of the sphere, so we can write h_cone = 2r_sphere.
Plugging these values into the formula for the volume of the cone, we have:
V_cone = (1/3)πr^2h
= (1/3)π(r_sphere)^2(2r_sphere)
= (1/3)πr_sphere^2(2r_sphere)
= (2/3)πr_sphere^3
Since the volume of the sphere (V_sphere) is given as 72 m^3, we have:
(4/3)πr_sphere^3 = 72
Simplifying this equation, we have:
r_sphere^3 = (3/4)(72/π)
r_sphere^3 = 54/π
Taking the cube root of both sides, we get:
r_sphere = (54/π)^(1/3)
Substituting this value of r_sphere back into the formula for the volume of the cone, we have:
V_cone = (2/3)πr_sphere^3
= (2/3)π[(54/π)^(1/3)]^3
= (2/3)π(54/π)
= 36
Therefore, the volume of the cone is 36 m^3.
The ratio that describes the relationship between the formulas for the volume of a cone to the volume of a sphere is: (2/3)πh/4πr^3.
wrong
I apologize for the incorrect response. The correct ratio that describes the relationship between the formulas for the volume of a cone and the volume of a sphere is 1:3.
The volume of a cone is 27 cm3 . What is the volume of a cylinder that shares the same radius and height as the cone?(1 point)
cm3
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height. In this case, we know that the volume of the cone is 27 cm^3.
27 = (1/3)πr^2h
We also know that the height of the cone is equal to twice the radius, or h = 2r.
Substituting h = 2r into the equation, we have:
27 = (1/3)πr^2(2r)
27 = (2/3)πr^3
To find the volume of a cylinder with the same radius and height, we use the formula V =πr^2h.
V = πr^2(2r)
V = 2πr^3
Therefore, the volume of a cylinder that shares the same radius and height as the cone is 2πr^3 cm^3.
The volume of a sphere is 72 m3 . What is the volume of a cone if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?
Use the image to answer the question.
A 3-dimensional cone is placed on top of a 3-dimensional cylinder. One of the cylinder bases is common to the cone, with the same circumference.
A 3D modeler is designing a component of a castle in a video game. She places a cone on top of a cylinder so they line up and share the same radius. The cone and the cylinder share the same height. If the volume of the cone is 50 cm3 , what is the volume of the whole object?
(1 point)
cm3
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wrong try again
I apologize for the mistake. Let's try again.
The volume of the cone is given as 50 cm^3.
The volume of a cone is given by the formula V_cone = (1/3)πr^2h, where r is the radius and h is the height.
Since the cone and cylinder share the same radius and height, we can use the volume of the cone to find the volume of the cylinder.
Let's denote the radius as r and the height as h.
The volume of the cone is given as 50 cm^3:
V_cone = (1/3)πr^2h = 50
Since the height of the cone is the same as the height of the cylinder, we can replace h in terms of r:
h = 2r
Substituting this into the volume of the cone equation:
V_cone = (1/3)πr^2(2r) = 50
Simplifying:
(2/3)πr^3 = 50
Now, let's solve for r:
r^3 = (3/2)(50/π)
r^3 = 75/π
r = (75/π)^(1/3)
To find the volume of the cylinder, we use the formula V_cylinder = πr^2h:
V_cylinder = π((75/π)^(1/3))^2(2(75/π)^(1/3))
= π(75/π)^(2/3)(150/π)^(1/3)
= 150
Therefore, the volume of the whole object, which consists of the cone and the cylinder, is 150 cm^3.