Suppose a sphere, cylinder, and cone all share the same radius and 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?
a. 1 cylinder = 2 cones + 1 sphere
b. sphere = cylinder + cone
c. 3 cones = 1 cylinder
d. sphere - cone = cylinder
5 answers
d. sphere - cone = cylinder
the volumes are:
sphere 4/3 π r^3
cylinder: πr^2h = πr^2(2h) = 2πr^3
cone: 1/3 πr^2 h = 2/3 πr^3
so, B or D
sphere 4/3 π r^3
cylinder: πr^2h = πr^2(2h) = 2πr^3
cone: 1/3 πr^2 h = 2/3 πr^3
so, B or D
Apologies for the mistake in my previous response. The correct answer is:
b. sphere = cylinder + cone
The volume formula for a sphere is V = 4/3 πr^3, the volume formula for a cylinder is V = πr^2h = πr^2(2r) = 2πr^3, and the volume formula for a cone is V = 1/3 πr^2h = 1/3 πr^2(2r) = 2/3 πr^3.
Adding the volume of the cone and the volume of the cylinder together gives: 2πr^3 + 2/3 πr^3 = (6/3)πr^3 + (2/3)πr^3 = 8/3 πr^3.
Therefore, the volume of the sphere is equal to the sum of the volume of the cylinder and the volume of the cone: 4/3 πr^3 = 8/3 πr^3.
b. sphere = cylinder + cone
The volume formula for a sphere is V = 4/3 πr^3, the volume formula for a cylinder is V = πr^2h = πr^2(2r) = 2πr^3, and the volume formula for a cone is V = 1/3 πr^2h = 1/3 πr^2(2r) = 2/3 πr^3.
Adding the volume of the cone and the volume of the cylinder together gives: 2πr^3 + 2/3 πr^3 = (6/3)πr^3 + (2/3)πr^3 = 8/3 πr^3.
Therefore, the volume of the sphere is equal to the sum of the volume of the cylinder and the volume of the cone: 4/3 πr^3 = 8/3 πr^3.
my bad. C is true, but does not include the sphere.
What is also true is that cylinder = sphere+ cone
but that's not one of the choices.
What is also true is that cylinder = sphere+ cone
but that's not one of the choices.
You are correct, and I apologize for the confusion. In this scenario, the relationship between the volume formulas for the sphere, cylinder, and cone is:
c. Three cones have the same volume as one cylinder.
This means that the volume of the cylinder is equal to the sum of the volumes of three cones. Unfortunately, none of the given choices match this relationship.
c. Three cones have the same volume as one cylinder.
This means that the volume of the cylinder is equal to the sum of the volumes of three cones. Unfortunately, none of the given choices match this relationship.