Question

what is the formula for the volume of a cone given h = c?
○ $v = \frac{4}{3}\pi r^3$
○ $v = \frac{1}{3}\pi r^2c$
○ $v = \pi r^2c$
○ $v = 2\pi rc$
○ $v = \pi rl$

Explanation:

Step1: Recall Cone Volume Formula

The standard formula for the volume \( V \) 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: Substitute \( h = c \)

Given that \( h = c \), we substitute \( h \) with \( c \) in the formula. So the formula becomes \( V = \frac{1}{3}\pi r^2 c \).

Step3: Eliminate Other Options

• \( V = \frac{4}{3}\pi r^3 \) is the volume of a sphere, so it's incorrect.
• \( V = \pi r^2 c \) is the volume of a cylinder (since cylinder volume is \( \pi r^2 h \), here \( h = c \) but without the \( \frac{1}{3} \) factor), so incorrect.
• \( V = 2\pi r c \) is related to the lateral surface area of a cylinder or circumference - related, not volume of a cone, incorrect.
• \( V=\pi rl \) is related to the lateral surface area of a cone (where \( l \) is slant height), not volume, incorrect.

Answer:

B. \( V = \frac{1}{3}\pi r^2 c \) (assuming the second option is labeled B, if we follow the order of options: first option A, second B, etc. The second option is \( V = \frac{1}{3}\pi r^2 c \))