Question
Suppose a cylinder and a cone share the same radius and the same height, which is twice the radius. Which ratio describes the relationship between the formulas for the volume of a cone to the volume of a cylinder?(1 point)
Sure, here is a step-by-step explanation:
1. Let's start by finding the formulas for the volume of a cone and the volume of a cylinder.
2. The formula for the volume of a cone is V_cone = (1/3) * π * r^2 * h, where r is the radius and h is the height.
3. The formula for the volume of a cylinder is V_cylinder = π * r^2 * h, where r is the radius and h is the height.
4. In this problem, we are given that the radius and the height of both the cone and the cylinder are the same. Let's call this value 'r'.
5. We are also given that the height of both the cone and the cylinder is twice the radius. So, the height 'h' can be written as 2r.
6. Substituting these values into the formulas for the volume of a cone and the volume of a cylinder:
- For the cone: V_cone = (1/3) * π * r^2 * (2r) = (2/3) * π * r^3
- For the cylinder: V_cylinder = π * r^2 * (2r) = 2π * r^3
7. Simplifying the expressions further:
- V_cone = (2/3) * π * r^3
- V_cylinder = 2π * r^3
8. Now, we can see that the ratio of the volume of a cone to the volume of a cylinder is:
- V_cone : V_cylinder = (2/3) * π * r^3 : 2π * r^3
9. Simplifying the ratio:
- V_cone : V_cylinder = (2/3) : 2
10. Further simplification:
- V_cone : V_cylinder = 1 : 3
Thus, the ratio that describes the relationship between the volume of a cone and the volume of a cylinder in this case is 1:3.
To find the ratio between the volume of a cone and the volume of a sphere in this case, we need to compare their formulas.
1. The formula for the volume of a cone is V_cone = (1/3) * π * r^2 * h, where r is the radius and h is the height.
2. The formula for the volume of a sphere is V_sphere = (4/3) * π * r^3, where r is the radius.
3. In this problem, we are given that the radius of the cone and the sphere is the same. Let's call this value 'r'.
4. We are also given that the height of the cone is twice the radius. So, the height 'h' of the cone can be written as 2r.
5. Substituting these values into the formulas for the volume of a cone and the volume of a sphere:
- For the cone: V_cone = (1/3) * π * r^2 * (2r) = (2/3) * π * r^3
- For the sphere: V_sphere = (4/3) * π * r^3
6. Simplifying the expressions further:
- V_cone = (2/3) * π * r^3
- V_sphere = (4/3) * π * r^3
7. Now, we can see that the ratio of the volume of a cone to the volume of a sphere is:
- V_cone : V_sphere = (2/3) * π * r^3 : (4/3) * π * r^3
8. Simplifying the ratio:
- V_cone : V_sphere = (2/3) : (4/3)
9. Further simplification:
- V_cone : V_sphere = 2 : 4
- The ratio reduces to 1 : 2
Therefore, the ratio that describes the relationship between the volume of a cone and the volume of a sphere in this case is 1 : 2.
To find the volume of the cylinder that shares the same radius and height as the cone, we can use the formula for the volume of a cone.
1. The formula for the volume of a cone is V_cone = (1/3) * π * r^2 * h, where r is the radius and h is the height.
2. In this problem, we are given that the volume of the cone is 27 cm³.
3. Let's substitute this value into the formula:
27 cm³ = (1/3) * π * r^2 * h
4. Since the cone and the cylinder share the same radius and height, we can replace 'r' and 'h' in the formula for the cone with the corresponding values for the cylinder.
V_cylinder = π * r^2 * h
5. Now we need to solve for the volume of the cylinder, V_cylinder.
From the previous step, we can see that V_cylinder = 3 * V_cone.
6. Substituting the value of V_cone = 27 cm³ into the equation, we get:
V_cylinder = 3 * 27 cm³ = 81 cm³
Therefore, the volume of the cylinder is 81 cm³.
To find the volume of a cone when the radius is the same as the sphere's radius and the height is equal to the sphere's diameter, we need to perform the following steps.
1. The formula for the volume of a sphere is V_sphere = (4/3) * π * r^3, where r is the radius.
2. In this problem, we are given that the volume of the sphere is 72 m³.
3. Let's substitute this value into the formula for the sphere:
72 m³ = (4/3) * π * r^3
4. Now, let's solve for the radius, r³:
r³ = (3/4) * (72 m³) / π
5. Simplify the expression:
r³ = 54 m³ / π
6. Take the cube root of both sides to find the radius, r:
r = (54 m³ / π)^(1/3)
7. Now, let's find the diameter of the sphere, which is twice the radius:
Diameter = 2 * r = 2 * (54 m³ / π)^(1/3)
8. Since the height of the cone is equal to the sphere's diameter, the height of the cone is:
Height = Diameter = 2 * (54 m³ / π)^(1/3)
9. Now, we have the radius and height of the cone. We can calculate its volume using the formula for the volume of a cone:
V_cone = (1/3) * π * r^2 * h
V_cone = (1/3) * π * (r^2) * (2 * (54 m³ / π)^(1/3))
10. Simplify the expression:
V_cone = (2/3) * π * (r^2) * (54 m³ / π)^(1/3)
= (2/3) * r^2 * (54 m³ / π)^(1/3)
11. Substitute the value of r:
V_cone = (2/3) * (54 m³ / π)^(2/3) * (54 m³ / π)^(1/3)
12. Combine the exponents:
V_cone = (2/3) * (54 m³ / π)^((2/3) + (1/3))
V_cone = (2/3) * (54 m³ / π)^(3/3)
V_cone = (2/3) * (54 m³ / π)
13. Calculate the value:
V_cone = 36 m³ / π
Therefore, the volume of the cone is 36 m³ / π.
The ratio that describes the relationship between the formulas for the volume of a cone and the volume of a cylinder in this case is 1:3.
step by step bot?
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)
:
To find the volume of the whole object, we need to sum up the volumes of the cone and the cylinder.
1. We are given that the volume of the cone is 50 cm³.
2. Let's say the radius of the cone and the cylinder is 'r' and the height is 'h', as they share the same dimensions.
3. The formula for the volume of a cone is V_cone = (1/3) * π * r^2 * h.
4. The volume of the cone is given as 50 cm³. So, we have:
50 cm³ = (1/3) * π * r^2 * h
5. Now, let's find the volume of the cylinder. The formula for the volume of a cylinder is V_cylinder = π * r^2 * h.
6. Since the cylinder and the cone share the same radius, the volume of the cylinder would be:
V_cylinder = π * r^2 * h.
7. To find the volume of the whole object, we simply add the volumes of the cone and the cylinder:
V_whole_object = V_cone + V_cylinder.
8. Substituting the values we know:
V_whole_object = 50 cm³ + (π * r^2 * h).
9. Since the height of the cone and the cylinder is the same, we can simplify this to:
V_whole_object = 50 cm³ + (π * r^2 * h).
10. Therefore, the volume of the whole object cannot be determined without knowing the values of 'r' and 'h'.