Question

find the volume of a cone with a base diameter of 9 m and a height of 6 m. write the exact volume in terms of π, and be sure to include the correct unit in your answer.

Explanation:

Step1: Encontrar el radio del cono

El diámetro es 9 m, así que el radio \( r = \frac{9}{2} = 4.5 \) m.

Step2: Usar la fórmula del volumen del cono

La fórmula del volumen de un cono es \( V = \frac{1}{3}\pi r^2 h \), donde \( h = 6 \) m. Sustituimos \( r = 4.5 \) y \( h = 6 \):
\( V = \frac{1}{3}\pi (4.5)^2 \times 6 \)
Calculamos \( (4.5)^2 = 20.25 \), entonces:
\( V = \frac{1}{3}\pi \times 20.25 \times 6 \)
Simplificamos: \( \frac{1}{3} \times 6 = 2 \), así que \( V = 2 \times 20.25 \pi = 40.5 \pi \)
Otra forma: \( 4.5 = \frac{9}{2} \), entonces \( r = \frac{9}{2} \), \( r^2 = \frac{81}{4} \)
\( V = \frac{1}{3}\pi \times \frac{81}{4} \times 6 = \frac{1}{3} \times 6 \times \frac{81}{4}\pi = 2 \times \frac{81}{4}\pi = \frac{81}{2}\pi = 40.5\pi \) (o \( \frac{81}{2}\pi \))

Answer:

\( \frac{81}{2}\pi \) metros cúbicos (o \( 40.5\pi \, \text{m}^3 \))