Which of the following is true about the relationship between the volume formulas for spheres, cylinders, and cones?
A. 1/3 cylinder = 1 cone
B. sphere = cylinder + cone
C. sphere - cone = cylinder
D. 1 cylinder = 2 cones + 1 sphere
3 answers
The correct answer is D. 1 cylinder = 2 cones + 1 sphere.
explain
The volume formulas for spheres, cylinders, and cones are as follows:
Sphere: (4/3)πr^3
Cylinder: πr^2h
Cone: (1/3)πr^2h
To compare the volumes, we need to find a relationship that allows us to equate the volumes of these shapes.
Option A suggests that 1/3 of a cylinder equals 1 cone. However, this is not true as the volume formula for a cone has a constant factor of 1/3, whereas the volume formula for a cylinder does not.
Option B suggests that the volume of a sphere is equal to the sum of the volumes of a cylinder and a cone. This is also incorrect since the volume formula for a cone involves the height, which is not present in the formula for a sphere.
Option C suggests that the difference between the volume of a sphere and a cone equals the volume of a cylinder. This is not correct either since the volume of a cone is 1/3 the volume of a cylinder, not equal to it.
Option D states that 1 cylinder is equal to 2 cones plus 1 sphere. This is the correct relationship. If we add twice the volume of a cone (2 × 1/3πr^2h) to the volume of a sphere (4/3πr^3), we will get the volume of a cylinder (πr^2h).
In conclusion, the correct answer is D. 1 cylinder = 2 cones + 1 sphere.
Sphere: (4/3)πr^3
Cylinder: πr^2h
Cone: (1/3)πr^2h
To compare the volumes, we need to find a relationship that allows us to equate the volumes of these shapes.
Option A suggests that 1/3 of a cylinder equals 1 cone. However, this is not true as the volume formula for a cone has a constant factor of 1/3, whereas the volume formula for a cylinder does not.
Option B suggests that the volume of a sphere is equal to the sum of the volumes of a cylinder and a cone. This is also incorrect since the volume formula for a cone involves the height, which is not present in the formula for a sphere.
Option C suggests that the difference between the volume of a sphere and a cone equals the volume of a cylinder. This is not correct either since the volume of a cone is 1/3 the volume of a cylinder, not equal to it.
Option D states that 1 cylinder is equal to 2 cones plus 1 sphere. This is the correct relationship. If we add twice the volume of a cone (2 × 1/3πr^2h) to the volume of a sphere (4/3πr^3), we will get the volume of a cylinder (πr^2h).
In conclusion, the correct answer is D. 1 cylinder = 2 cones + 1 sphere.
1 answer