Question

The volume of a sphere whose diameter is 18 centimeters is _____ cubic centimeters. If its diameter were reduced by half, its volume would be _____ of its original volume.

a) 3052.4, one-half
b) 381.0, one-fourth
c) 3052.4, one-fourth
d) 381.0, one-half

Answer 1

The volume of a sphere with an 18 centimeter diameter is 3053.6 cm³. When the diameter is halved, the volume becomes 381.7 cm³, which is one-eighth of the original volume. The closest correct answer is option c, assuming there was a typo for 'one-eighth'.

Step-by-step explanation:

The question asks about the volume of a sphere with a given diameter and how the volume changes when the diameter is reduced by half. To solve for the volume of a sphere, we use the formula V = (4/3)πr³, where r is the radius of the sphere. Given that the diameter of the sphere is 18 centimeters, the radius would be half of that, i.e., 9 centimeters. The volume is therefore calculated as follows:

V = (4/3)π(9 cm)³ = (4/3)π(729 cm³) ≈ 3053.6 cm³.

If the diameter is reduced by half, the new diameter will be 9 centimeters, and the radius will then be 4.5 centimeters. The new volume is then:

V = (4/3)π(4.5 cm)³ = (4/3)π(91.125 cm³) ≈ 381.7 cm³.

This new volume is one-eighth of the original volume because the volume of the sphere depends on the cube of the radius, and (1/2)³ is 1/8. Therefore, option c (3052.4 cm³, one-fourth) is incorrect, as the correct answer should be d (381.0 cm³, one-eighth). However, since 'one-eighth' is not given as an option and assuming it is a typo, 'one-fourth' being the closest to the actual answer, makes option d the best choice.