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. 3 cones + 1 cylinder
2. sphere= Cylinder + cone
3. sphere - cone= cylinder
4. 1 cylinder = 2 cones + 1 sphere
The correct option is:
4. 1 cylinder = 2 cones + 1 sphere
Can you explain why this is the correct answer
Certainly!
Let's compare the volume formulas for the sphere, cylinder, and cone.
The volume of a sphere with radius r is given by the formula: V_sphere = (4/3)πr^3.
The volume of a cylinder with radius r and height h is given by the formula: V_cylinder = πr^2h.
The volume of a cone with radius r and height h is given by the formula: V_cone = (1/3)πr^2h.
Given that the height of both the cylinder and cone is twice the radius, we can substitute h = 2r into the volume formulas.
Substituting h = 2r into the volume formulas for the cylinder and cone, we get:
V_cylinder = πr^2(2r) = 2πr^3,
V_cone = (1/3)πr^2(2r) = (2/3)πr^3.
Using these revised volume formulas, let's compare the volumes of 1 cylinder, 2 cones, and 1 sphere.
1 cylinder = 2πr^3,
2 cones = 2(2/3)πr^3 = (4/3)πr^3,
1 sphere = (4/3)πr^3.
We can see that the volume of 1 cylinder is equal to the sum of the volumes of 2 cones and 1 sphere. Therefore, the correct option is 4. 1 cylinder = 2 cones + 1 sphere.
Thank you r amazing at your job
You're welcome! I'm glad I could help. If you have any more questions, feel free to ask!
To find the relationship between the volume formulas for the sphere, cylinder, and cone, we need to understand the formulas for each shape.
1. The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius.
2. The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
3. 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.
Now, let's analyze the given information:
- The sphere, cylinder, and cone all share the same radius.
- The cylinder and cone have the same height, which is twice the radius.
Since the radius is the same for all three shapes, we can use the same variable r.
- The height of the cylinder and cone is 2r.
- The volume of the sphere is (4/3)πr^3.
- The volume of the cylinder is πr^2(2r) = 2πr^3.
- The volume of the cone is (1/3)πr^2(2r) = (2/3)πr^3.
Now, let's compare the volume formulas based on the information above:
1. The volume of 3 cones is (3 * ((2/3)πr^3)) = (2πr^3). The volume of 1 cylinder is πr^2(2r) = 2πr^3. So, 3 cones + 1 cylinder is equal to the volume of the sphere. Therefore, option 1 is correct.
2. The volume of the cylinder and the volume of the cone equal (2πr^3 + (2/3)πr^3) = (8/3)πr^3. This is not equal to the volume of the sphere. Therefore, option 2 is incorrect.
3. The volume of the sphere minus the volume of the cone is ((4/3)πr^3 - (2/3)πr^3) = (2/3)πr^3. This is not equal to the volume of the cylinder. Therefore, option 3 is incorrect.
4. The volume of 2 cones plus the volume of 1 sphere is (2 * ((2/3)πr^3)) + (4/3)πr^3 = (8/3)πr^3. This is not equal to the volume of the cylinder. Therefore, option 4 is incorrect.
Therefore, the correct answer is option 1: 3 cones + 1 cylinder.