Question

which of the following dimensions do not affect the volume of a cone?
a. radius of the base
b. diameter of the base
c. slant height of the cone
d. height of the cone

Explanation:

Step1: Recall the volume formula of a cone

The volume formula of a cone is \(V=\frac{1}{3}\pi r^{2}h\), where \(r\) is the radius of the base and \(h\) is the height of the cone.

Step2: Analyze the relationship between diameter and radius

Since \(d = 2r\) (where \(d\) is the diameter of the base), the diameter is related to the radius. Substituting \(r=\frac{d}{2}\) into the volume formula, we get \(V=\frac{1}{3}\pi(\frac{d}{2})^{2}h=\frac{1}{12}\pi d^{2}h\). So the diameter affects the volume.

Step3: Analyze the relationship between slant height and volume

The slant height \(l\) of a cone is related to the radius \(r\) and height \(h\) by the formula \(l=\sqrt{r^{2}+h^{2}}\). But the volume formula \(V=\frac{1}{3}\pi r^{2}h\) does not contain the slant height \(l\) directly. Changing the slant height while keeping \(r\) and \(h\) constant does not change the volume.

Answer:

C. Slant height of the cone