Use the relationship among the formulas for the volume of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 cubic cm. What is the volume of a sphere if its radius is the same as the cylinders and the height of the cylinder is equal to the sphere's diameter
We can start by finding the radius of the cylinder. Since we only know the volume of the cylinder, we can use the formula for the volume of a cylinder:
Volume of a cylinder = π × radius² × height
We are given that the volume of the cylinder is 36 cubic cm, so we can set up the equation:
36 = π × radius² × height
Since the height of the cylinder is equal to the diameter of the sphere, we can substitute the height with 2 × radius:
36 = π × radius² × (2 × radius)
Simplifying this equation, we have:
36 = 2π × radius³
Now, let's determine the radius:
36 / (2π) = radius³
18 / π = radius³
radius = (18 / π)^(1/3)
Now that we have the radius of the cylinder, we can use the formula for the volume of a sphere to find the volume of the sphere:
Volume of a sphere = (4/3) × π × radius³
Substituting the value of radius that we found earlier, we get:
Volume of a sphere = (4/3) × π × [(18 / π)^(1/3)]³
Simplifying further, we have:
Volume of a sphere = (4/3) × π × [18 / π]^(1/3) × [18 / π]^(1/3) × [18 / π]^(1/3)
Volume of a sphere = (4/3) × π × (18 / π)^(1/3)³
Volume of a sphere = (4/3) × π × (18 / π)
Volume of a sphere = 4π × (9 / π)
Volume of a sphere = 4 × 9
Volume of a sphere = 36 cubic cm
Therefore, the volume of the sphere with the same radius as the cylinder and a height equal to the diameter of the sphere is 36 cubic cm.
redo your answer
Apologies for the confusion in the previous response. Let's go through the problem again:
Given: Volume of a cylinder = 36 cubic cm
We know that the volume of a cylinder is given by the formula:
Volume of a cylinder = π × radius² × height
Since the height of the cylinder is equal to the diameter of the sphere, we can represent it as 2r (where r is the radius of the sphere).
So, the equation for the volume of the cylinder becomes:
36 = π × radius² × 2r
Simplifying further:
36 = 2πr³
Now, let's solve for the radius of the cylinder:
r³ = 36 / (2π)
r = (36 / (2π))^(1/3)
Now that we have the radius of the cylinder, we can use it to find the volume of the sphere. The formula for the volume of a sphere is:
Volume of a sphere = (4/3) × π × radius³
Substituting the value of the radius we found earlier:
Volume of a sphere = (4/3) × π × [(36 / (2π))^(1/3)]³
Simplifying further:
Volume of a sphere = (4/3) × π × (36 / (2π))^(1/3)³
Volume of a sphere = (4/3) × π × (36 / (2π))^1
Volume of a sphere = (4/3) × π × (18 / π)
Volume of a sphere = 4 × 3 × 18
Volume of a sphere = 216 cubic cm
Therefore, the volume of the sphere with the same radius as the cylinder and a height equal to the diameter of the sphere is 216 cubic cm.
Use the relationship among the formulas for the volume of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 cubic cm. What is the volume of a sphere if its radius is the same as the cylinders and the height of the cylinder is equal to the sphere's diameter
A. 24 cubic cm
B. 54 cubic cm
C. 12 cubic cm
D. 18 cubic cm
To solve this problem, we need to find the volume of a sphere when its radius is the same as the cylinder and the height of the cylinder is equal to the sphere's diameter.
Given:
Volume of the cylinder = 36 cubic cm
We know that the formula for the volume of a cylinder is:
Volume of a cylinder = π × radius² × height
Since the height of the cylinder is equal to the diameter of the sphere, we can represent it as 2r (where r is the radius of the sphere).
So, the equation for the volume of the cylinder becomes:
36 = π × radius² × 2r
Simplifying further:
36 = 2πr³
Now, let's solve for the radius of the cylinder:
r³ = 36 / (2π)
r = (36 / (2π))^(1/3)
Now that we have the radius of the cylinder, we can use it to find the volume of the sphere. The formula for the volume of a sphere is:
Volume of a sphere = (4/3) × π × radius³
Substituting the value of the radius we found earlier:
Volume of a sphere = (4/3) × π × [(36 / (2π))^(1/3)]³
Simplifying further:
Volume of a sphere = (4/3) × π × (36 / (2π))^(1/3)³
Volume of a sphere = (4/3) × π × (36 / (2π))^1
Volume of a sphere = (4/3) × π × (18 / π)
Volume of a sphere = 4 × 3 × 18
Volume of a sphere = 216 cubic cm
Therefore, the volume of the sphere when its radius is the same as the cylinder's and the height of the cylinder is equal to the sphere's diameter is 216 cubic cm.
So, the correct answer is not provided in the options given.
choose an answer
Apologies for the previous incorrect response. Let's solve the problem again:
Given:
Volume of the cylinder = 36 cubic cm
We know that the formula for the volume of a cylinder is:
Volume of a cylinder = π × radius² × height
Since the height of the cylinder is equal to the diameter of the sphere, we can represent it as 2r (where r is the radius of the sphere).
So, the equation for the volume of the cylinder becomes:
36 = π × radius² × 2r
Simplifying further:
36 = 2πr³
Now, let's solve for the radius of the cylinder:
r³ = 36 / (2π)
r = (36 / (2π))^(1/3)
Now that we have the radius of the cylinder, we can use it to find the volume of the sphere. The formula for the volume of a sphere is:
Volume of a sphere = (4/3) × π × radius³
Substituting the value of the radius we found earlier:
Volume of a sphere = (4/3) × π × [(36 / (2π))^(1/3)]³
Simplifying further:
Volume of a sphere = (4/3) × π × (36 / (2π))^(1/3)³
Volume of a sphere = (4/3) × π × (36 / (2π))^1
Volume of a sphere = (4/3) × π × (18 / π)
Volume of a sphere = 4 × 3 × 18
Volume of a sphere = 216 cubic cm
Therefore, the volume of the sphere when its radius is the same as the cylinder's and the height of the cylinder is equal to the sphere's diameter is 216 cubic cm.
Since none of the options provided matches the calculated volume of 216 cubic cm, it appears there is no correct answer among the options.