Question

note: image is not drawn to scale. what is the volume of this right cone? 27π cm³ 200π cm³ 213π cm³ 300π cm³

Explanation:

Step1: Recall the volume formula for a cone

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

Step2: Identify the values of \( r \) and \( h \)

From the diagram, the radius \( r = 10 \) cm (wait, no, wait. Wait, the dashed line to the side is 10 cm? Wait, no, the height is 9 cm? Wait, no, let's check again. Wait, the right triangle: the slant height is 13.5 cm, height is 9 cm? Wait, no, the radius is the horizontal dashed line. Wait, maybe I misread. Wait, the diagram: the height (vertical dashed line) is 9 cm? Wait, no, the horizontal dashed line (radius) is 10 cm? Wait, no, wait, the formula: \( V = \frac{1}{3}\pi r^2 h \). Wait, maybe the height is 9 cm? Wait, no, let's check the options. Wait, maybe the radius is 10? Wait, no, let's recalculate. Wait, maybe the height is 9? Wait, no, let's see:

Wait, the vertical dashed line (height) is 9 cm? Wait, no, the horizontal dashed line (radius) is 10 cm? Wait, no, maybe the height is 9, radius is 10? Wait, no, let's compute:

Wait, \( V = \frac{1}{3}\pi r^2 h \). Let's plug in \( r = 10 \), \( h = 9 \):

\( V = \frac{1}{3}\pi (10)^2 (9) = \frac{1}{3}\pi \times 100 \times 9 = 300\pi \)? No, that's 300π, but that's one of the options. Wait, but wait, maybe the height is 9, radius is 10? Wait, but let's check the slant height: slant height \( l = \sqrt{r^2 + h^2} \). If \( r = 10 \), \( h = 9 \), then \( l = \sqrt{10^2 + 9^2} = \sqrt{100 + 81} = \sqrt{181} \approx 13.45 \), which is close to 13.5 cm. So that matches. So radius \( r = 10 \) cm, height \( h = 9 \) cm.

So plug into the formula:

\( V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (10)^2 (9) \)

Step3: Calculate the volume

First, compute \( (10)^2 = 100 \). Then, \( 100 \times 9 = 900 \). Then, \( \frac{1}{3} \times 900 = 300 \). So \( V = 300\pi \) \( \text{cm}^3 \). Wait, but that's one of the options: 300π cm³. Wait, but let's check again. Wait, maybe I made a mistake. Wait, the height is 9, radius is 10. Then \( \frac{1}{3} \times \pi \times 10^2 \times 9 = \frac{1}{3} \times \pi \times 100 \times 9 = 300\pi \). So that's 300π cm³.

Answer:

\( 300\pi \text{ cm}^3 \) (the option with 300π cm³)