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
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.
To determine which of the given options is true about the relationship between the volume formulas for spheres, cylinders, and cones, let's examine each choice individually.
A. 1/3 cylinder = 1 cone: This option suggests that the volume of 1/3 of a cylinder is equal to the volume of 1 cone. However, this is not correct. The formula for the volume of a cone is (1/3)πr²h, which is different from the formula for the volume of a cylinder, πr²h. Therefore, option A is not true.
B. sphere = cylinder + cone: This option suggests that the volume of a sphere is equal to the volume of a cylinder plus the volume of a cone. Similarly to option A, this is false. The formula for the volume of a sphere is (4/3)πr³, which is distinct from both the cylinder and cone formulas. Hence, option B is not true.
C. sphere - cone = cylinder: This option suggests that subtracting the volume of a cone from the volume of a sphere would yield the volume of a cylinder. However, this is also incorrect. As stated before, the formulas for the volumes of spheres, cylinders, and cones are different, so there is no straightforward relationship between these volumes. Therefore, option C is not true.
D. 1 cylinder = 2 cones + 1 sphere: This option suggests that the volume of 1 cylinder is equal to the volume of 2 cones plus the volume of 1 sphere. Remarkably, this option is also false. The volume formulas for cylinders, cones, and spheres are distinct from one another, so there is no direct relationship as described in the option. Thus, option D is not true.
In conclusion, none of the given options (A, B, C, or D) are true about the relationship between the volume formulas for spheres, cylinders, and cones.