To solve this problem, we need to know the relationship between the volume of a cylinder, cone, and sphere.
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
The volume of a cone is given by the formula V = (1/3)πr^2h.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius.
In this problem, the height of the cylinder is equal to the sphere's diameter. We know the volume of the cylinder is 36 cm^3.
Let's use the information given to find the radius and height of the cylinder. We can then use these values to find the volume of the sphere.
Given that the volume of the cylinder is 36 cm^3, we can use the formula V = πr^2h to solve for the height. So, 36 = πr^2h.
Since the height is equal to the diameter of the sphere, we can write h = 2r.
Substituting h = 2r into the equation, we have 36 = πr^2(2r), which simplifies to 36 = 2πr^3.
Dividing both sides by 2π, we get r^3 = 18/π.
Taking the cube root of both sides, we have r = (18/π)^(1/3).
Now that we know the radius of the sphere, we can substitute this value into the formula for the volume of a sphere to find the volume of the sphere.
V = (4/3)πr^3
V = (4/3)π((18/π)^(1/3))^3
V = (4/3)π((18/π)^(1/3))^(3/1)
V = (4/3)π(18/π)
V = (4/3)(18)
V = 24 cm^3
Therefore, the volume of the sphere with the same radius as the cylinder and height equal to the sphere's diameter is 24 cm^3.
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)
5 answers
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)
The relationship between the volume formulas for the sphere, cylinder, and cone is as follows:
The volume of the sphere is given by the formula V = (4/3)πr^3, where r is the radius.
The volume of the cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
The volume of the cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.
Given that the radius is the same for all three shapes, we can compare the volume formulas.
First, let's compare the formulas for the cylinder and cone:
The volume of the cylinder is V = πr^2h.
The volume of the cone is V = (1/3)πr^2h.
We can see that the volume of the cone is one-third the volume of the cylinder. This is because the factor of 1/3 in the volume formula for the cone accounts for the cone's shape, which narrows down towards the top.
Next, let's compare the formulas for the sphere and the cylinder:
The volume of the sphere is V = (4/3)πr^3.
The volume of the cylinder is V = πr^2h.
We can't directly compare these two formulas because they have different variables - the sphere's volume formula includes r^3, while the cylinder's volume formula includes r^2h.
Therefore, we cannot determine a specific relationship between the volume formulas for the sphere, cylinder, and cone based on the information given.
The volume of the sphere is given by the formula V = (4/3)πr^3, where r is the radius.
The volume of the cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
The volume of the cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.
Given that the radius is the same for all three shapes, we can compare the volume formulas.
First, let's compare the formulas for the cylinder and cone:
The volume of the cylinder is V = πr^2h.
The volume of the cone is V = (1/3)πr^2h.
We can see that the volume of the cone is one-third the volume of the cylinder. This is because the factor of 1/3 in the volume formula for the cone accounts for the cone's shape, which narrows down towards the top.
Next, let's compare the formulas for the sphere and the cylinder:
The volume of the sphere is V = (4/3)πr^3.
The volume of the cylinder is V = πr^2h.
We can't directly compare these two formulas because they have different variables - the sphere's volume formula includes r^3, while the cylinder's volume formula includes r^2h.
Therefore, we cannot determine a specific relationship between the volume formulas for the sphere, cylinder, and cone based on the information given.
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 cylinder = sphere – cone cylinder = sphere – cone cone = sphere – cylinder cone = sphere – cylinder cone = cylinder – sphere cone = cylinder – sphere sphere = cylinder + cone
Based on the given information, the correct relationship between the volume formulas for the sphere, cylinder, and cone is:
cylinder = sphere – cone
cylinder = sphere – cone