Question

A cone has a height of `24 cm` and its base has a radius of `9 cm`. If the cone is horizontally cut into two segments `16 cm` from the base, what would the surface area of the bottom segment be?

Answer 1

Total Surface Area of the truncated cone `color(red)(A_T = 926.986)` `cm^3`

Explanation:

Given :
`H = 24, h = 24 - 16 = 8, R = 9`

`r = (h / H) * R = (8/24) * 9 = 3`

Lateral Surface Area of a cone `A_l = pi r l = pi r sqrt(h^2 + r^2)`

L S A of uncut cone `A_L = pi * R * L = pi * R * sqrt(H^2 + R^2)`

`A_L = pi * 9;* sqrt(24^2 + 9^2) = color(blue)(724.728)`

Similarly, L S A of cut cone is

`A_l = pi * r * l = pi * 3 * sqrt (8^2 + 3^2) = color(blue)(80.5253)`

L S A of truncated cone `A_(tc) = A_L - A_l = 724.728 - 80.5253 = color (green)(644.2427)`

Area of base of uncut cone `A_B = pi R^2 = pi * 9^2 = color (green)(254.469)`

Area of base of cut cone `A_b = pi r^2 = pi * 3^2 = color(green)(28.2743)`

Total Surface Area of the truncated cone

`A_T = A_L - A_l + A_B + A_b = 644.2427 + 254.469 + 28.2743`

`A_T = color (brown)(926.986)` `color(purple)(cm^3)`