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?

1 Answer
Jan 24, 2018

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

Explanation:

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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)#