Question

A cone has a height of `36 cm` and its base has a radius of `15 cm`. If the cone is horizontally cut into two segments `24 cm` from the base, what would the surface area of the bottom segment be?

Answer 1

Total surface area of bottom segment `=2419.0263`

Explanation:

Slanting length (`l_1`) of uncut cone `= sqrt(36^2 + 15^2`
`l_1 = sqrt1521 = 39 cm`

Lateral surface area of uncut cone `= pi r l`
`= pi* 15 * 39 = 1837.8317 cm^2`

Slanting length of cut cone with height 12 cm is
`l_2`, radius `r_2`
`l_2/l_1 = h_2/h_1 = r_2/r_1`
`l_2/39 = 12/36 = r_2/15`
`l_2 = (39*12)/36 = 13 cm.`
`r_2 = (15*12)/36 = 5 cm`
Lateral surface area of cut cone `= pi*r_2 * l_2`
`= pi* 5 * 13 = 204.2035 cm^2`

Lateral surface area of cut base `= (pi*r_1*l_1) - (pi*r_2*l_2)`
`= 1837.8317 - 204.2035 = 1633.6282 cm^2`, (1)

Area of uncut cone base `= pi*r_1^2 = pi*15^2 = 706.8583 cm^2`, (2)

Area of cut cone base`= pi*r_2^2 = pi*5^2 = 78.5398 cm^2`, (3)

Total surface area of cut base `= (1) + (2) + (3)`
`= 1633.6282+706.8583+78.5398 = 2419.0263 cm^2`