Question

A stalactite is a mineral formation that hangs from the ceiling of a cave. A cone shaped stalactite has a height of 48 centimeters and a base circumference of \(3.5\pi \) centimeters. What is the volume of the stalactite?

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the volume of the cone-shaped stalactite, we can use the formula for the volume of a cone:

\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone.

First, we need to find the radius of the base. The circumference of the base is given as \( 3.5\pi \) centimeters, so we can find the radius using the formula for the circumference of a circle:
\[ C = 2\pi r \]
\[ 3.5\pi = 2\pi r \]
\[ r = \frac{3.5\pi}{2\pi} = 1.75 \text{ cm} \]

Now we can plug in the values of the radius and height into the formula for the volume of a cone:
\[ V = \frac{1}{3} \pi (1.75)^2 \times 48 \]
\[ V = \frac{1}{3} \pi \times 3.0625 \times 48 \]
\[ V = \frac{1}{3} \pi \times 146.75 \]
\[ V = 48.917 \pi \approx 153.62 \text{ cm}^3 \]

Therefore, the volume of the cone-shaped stalactite is approximately 153.62 cubic centimeters.