The correct option is:
4. 1 cylinder = 2 cones + 1 sphere
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
5 answers
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.
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!