Use the relationship among the formulas for the volumes of cones, cylinders, and sphere to solve the following problem. The volume of a cylinder is 36 cm^3. 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. 24 cm^3
2. 54 cm^3
3. 12 cm^3
4. 18 cm^3
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
The height of the cylinder is equal to the sphere's diameter, so the height of the cylinder is 2r.
The volume of the cylinder is given as 36 cm^3.
Therefore, we have 36 = πr^2(2r).
Simplifying this equation, we get 36 = 2πr^3.
Dividing both sides by 2π, we get 18 = r^3.
To find the volume of the sphere, we use the formula V = (4/3)πr^3.
Plugging in the value of r as 18, we get V = (4/3)π(18^3) = 12π(5832) = 69984π.
Approximating π as 3.14, we get 69984(3.14), which is approximately equal to 219725.76 cm^3.
The correct answer is not provided, as 219725.76 cm^3 is not one of the given choices.
what's the answer bot?
I apologize for the mistake in the previous response. Let's find the correct answer.
We know that the volume of a cylinder is given by V_cylinder = πr^2h, where r is the radius and h is the height.
The height of the cylinder is given as the diameter of the sphere, so h = 2r.
We are given that the volume of the cylinder is 36 cm^3.
Therefore, we have 36 = πr^2(2r).
Simplifying this equation, we get 36 = 2πr^3.
Dividing both sides by 2π, we get 18 = r^3.
To find the volume of the sphere, we use the formula V_sphere = (4/3)πr^3.
Plugging in the value of r as 18, we get V_sphere = (4/3)π(18^3) = (4/3)π(5832) = 7776π.
Approximating π as 3.14, we get 7776(3.14), which is approximately equal to 24416.64 cm^3.
Therefore, the volume of the sphere, when the radius is the same as the cylinder's and the height of the cylinder is equal to the sphere's diameter, is approximately 24416.64 cm^3.
The closest option is "2. 54 cm^3". However, the correct answer is not provided in the given choices.
To solve this problem, we need to use the relationship among the formulas for the volumes of cones, cylinders, and spheres.
The volume of a cylinder is given by the formula V_cylinder = π * r^2 * h, where r is the radius of the cylinder and h is the height.
The volume of a cone is given by the formula 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 the formula V_sphere = 4/3 * π * r^3, where r is the radius of the sphere.
In this problem, the volume of the cylinder is given as 36 cm^3. We are told that the radius of the sphere is the same as the radius of the cylinder, and the height of the cylinder is equal to the sphere's diameter.
Since the cylinder's volume is given as 36 cm^3, we can substitute the given values into the formula for the volume of a cylinder:
36 = π * r^2 * h
Since the height of the cylinder is equal to the sphere's diameter, we can write h = 2r.
Substituting this value into the equation, we get:
36 = π * r^2 * 2r
Simplifying the equation, we get:
36 = 2π * r^3
Dividing both sides of the equation by 2π, we get:
r^3 = 18
Taking the cube root of both sides of the equation, we get:
r = 3
Now that we know the radius of the sphere is 3 cm, we can substitute this value into the formula for the volume of a sphere:
V_sphere = 4/3 * π * (3)^3
Simplifying the equation, we get:
V_sphere = 4/3 * π * 27
V_sphere = 36π
Therefore, the volume of the sphere is 36π cm^3.
To determine the answer choice, we need to approximate the value of π. The most common approximation is 3.14, which we will use here.
Calculating 36 * 3.14, we get:
V_sphere ≈ 113.04 cm^3
None of the answer choices match 113.04 cm^3, so the correct approach would be to round our answer to the nearest option. The nearest option to 113.04 cm^3 is 112 cm^3, which is not provided in the answer choices.
Therefore, there is no exact answer provided among the options.