A cone has a height of #15 cm# and its base has a radius of #9 cm#. If the cone is horizontally cut into two segments #7 cm# from the base, what would the surface area of the bottom segment be?

Question

A cone has a height of #15  cm# and its base has a radius of #9  cm#.  If the cone is horizontally cut into two segments #7 cm# from the base, what would the surface area of the bottom segment be?

Aaliyah Anderson · Accepted Answer

Total Surface Area = 38.6979#cm^2#

Mason Adams · Answer

undefined To find the surface area of the bottom segment of the cone, you need to calculate the lateral surface area of the whole cone and subtract the lateral surface area of the top segment.

First, calculate the slant height of the cone using the Pythagorean theorem: 
$$l = \sqrt{r^2 + h^2}$$

$$l = \sqrt{9^2 + 15^2} = \sqrt{81 + 225} = \sqrt{306} \approx 17.49 \text{ cm}$$

Then, calculate the lateral surface area of the whole cone:
$$A_{\text{lateral}} = \pi \times r \times l$$
$$A_{\text{lateral}} = \pi \times 9 \times 17.49$$

Next, calculate the height of the top segment:
$$h_{\text{top}} = h - 7 = 15 - 7 = 8 \text{ cm}$$

Calculate the slant height of the top segment:
$$l_{\text{top}} = \sqrt{r^2 + h_{\text{top}}^2}$$
$$l_{\text{top}} = \sqrt{9^2 + 8^2} = \sqrt{81 + 64} = \sqrt{145} \approx 12.04 \text{ cm}$$

Then, calculate the lateral surface area of the top segment:
$$A_{\text{lateral, top}} = \pi \times r \times l_{\text{top}}$$
$$A_{\text{lateral, top}} = \pi \times 9 \times 12.04$$

Finally, subtract the lateral surface area of the top segment from the lateral surface area of the whole cone to get the surface area of the bottom segment:
$$A_{\text{bottom segment}} = A_{\text{lateral}} - A_{\text{lateral, top}}$$