Question

A cone has a height of `7 cm` and its base has a radius of `2 cm`. If the cone is horizontally cut into two segments `3 cm` from the base, what would the surface area of the bottom segment be?

Answer 1

Total Surface Area of the truncated cone is `color(green)(A_T = 47.4757`

Explanation:

`R / H = r / (H - h)`

`r = (R * (H-h)) / H = (2 * (7-3)) / 7 = 8/7`

Lateral surface area of full cone

`A_f = pi R * L_1 = pi R sqrt(R^2 + H^2)`

`A_f = pi 2 * sqrt(2^2 + 7^2) = 45.7423`

Lateral Surface Area of top porion of the cut cone

`A_c = pi r L_2 = pi * (8/7) * sqrt((8/7)^2 + (7-3)^2) = 14.9363`

Lateral Surface area of the truncated cone

`A_t = A_f - A_c = 45.7423 - 14.9363 = color(brown)(30.806`

Base area of full cone

`A_B = pi R^2 = pi * 2^2 = color(brown)(12.5664`

Base area of top portion of the cut cone (top surface area of the frustum of the cone)

`A_b = pi * r^2 = pi * (8/7)^2 = color(brown)(4.1033)`

Total Surface Area of the truncated cone is

`A_T = A_t + A_B + A_b = 30.806 + 12.5664 + 4.1033) = color(green)(47.4757)`