Question

which of the following formulas is used to calculate the area of the base of a cone?

a. \\( \pi r ^ { 2 } h \\)

b. \\( \frac { 1 } { 3 } \pi r ^ { 2 } \\)

c. \\( \frac { 4 } { 3 } \pi r ^ { 3 } \\)

d. \\( \pi r ^ { 2 } \\)

which of the following dimensions do not affect the volume of a cone?

a. height of the cone

b. slant height of the cone

c. radius of the base

d. diameter of the base

Explanation:

First Question:

Step1: Recall the formula for the area of a circle

The base of a cone is a circle. The formula for the area of a circle is \(A = \pi r^{2}\), where \(r\) is the radius of the circle.

• Option a: \(\pi r^{2}h\) is the formula for the volume of a cylinder.
• Option b: \(\frac{1}{3}\pi r^{2}\) is not a standard formula for a common geometric measure (not area of base, volume, etc. in a simple sense for basic 3 - D shapes related to cones).
• Option c: \(\frac{4}{3}\pi r^{3}\) is the formula for the volume of a sphere.
• Option d: \(\pi r^{2}\) is the formula for the area of a circle (base of the cone).

Second Question:

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 (\(d = 2r\) relates diameter \(d\) and radius \(r\)), and \(h\) is the height of the cone. The slant height \(l\) of a cone is related to \(r\) and \(h\) by the Pythagorean theorem \(l=\sqrt{r^{2}+h^{2}}\), but it does not appear in the volume formula \(V=\frac{1}{3}\pi r^{2}h\).

• Option a: Height \(h\) is in the volume formula \(V = \frac{1}{3}\pi r^{2}h\), so it affects the volume.
• Option b: Slant height \(l\) is not in the volume formula \(V=\frac{1}{3}\pi r^{2}h\), so it does not affect the volume.
• Option c: Radius \(r\) is in the volume formula \(V=\frac{1}{3}\pi r^{2}h\), so it affects the volume.
• Option d: Since \(d = 2r\) (and \(r=\frac{d}{2}\)), and \(r\) is in the volume formula \(V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi(\frac{d}{2})^{2}h=\frac{1}{12}\pi d^{2}h\), diameter affects the volume.

Answer:

For the first question: d. \(\pi r^{2}\)
For the second question: b. Slant height of the cone