Question

1. the volume of a sphere depends on the cube of its:

a. height
b. radius
c. circumference
d. diameter

Explanation:

Step1: Recall the volume formula of a sphere

The volume formula of a sphere is \(V = \frac{4}{3}\pi r^{3}\), where \(V\) is the volume and \(r\) is the radius.

Step2: Analyze the relationship between volume and radius

From the formula \(V=\frac{4}{3}\pi r^{3}\), we can see that the volume \(V\) is directly proportional to the cube of the radius \(r\) (\(V\propto r^{3}\)).

Step3: Analyze other options

• Option a: A sphere has no height in the traditional sense like a prism or cylinder.
• Option c: The circumference of a sphere \(C = 2\pi r\), and if we express \(r=\frac{C}{2\pi}\) and substitute into the volume formula \(V=\frac{4}{3}\pi(\frac{C}{2\pi})^{3}=\frac{C^{3}}{6\pi^{2}}\), which is more complex than the direct \(r^{3}\) relationship.
• Option d: The diameter \(d = 2r\), and \(V=\frac{4}{3}\pi(\frac{d}{2})^{3}=\frac{\pi d^{3}}{6}\). While the volume is also related to the cube of the diameter, the formula is more naturally expressed in terms of radius. Since the question asks for the most direct measure (as the standard formula is in terms of radius), radius is the better answer.

Answer:

B. Radius