A cone has a height of #9 cm# and its base has a radius of #4 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?

Question

A cone has a height of #9  cm# and its base has a radius of #4  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?

Mason Adams · Accepted Answer

#:.color(purple)(=141.36cm^2# to the nearest 2 decimal places # cm^2# :.Pythagoras: #c^2=9^2+4^2# #:.c=L=sqrt(9^2+4^2)# #:. c=Lcolor(purple)(=9.849cm# #:.9/4=tan theta=2.25=66^@02’15”# :.#"color(purple)(S.A".##=pi*r*L# :.S.A.#=pi*4*9.849# :.S.A.#=123.766# :.Total S.A.#color(purple)(=123.766cm^2# #:.Cot 66^@02’15”*6=2.667cm=#radius of top part :.Pythagoras: #c^2=6^2+2.667^2# #:.c=L=sqrt(6^2+2.667^2)# #:. c=Lcolor(purple)(=6.566cm# top part :.S.A. top part#=pi*r*L# S.A. top part#:.pi*2.667*6.566# S.A. top part#:.=55.014# S.A. top part#:.color(purple)(=55.014cm^2# :.S.A. Bottom part#color(purple)(=123.766-55.014=68.758cm^2# :.S.A. Bottom part#=68.758+pir^2+pir^2# #:.68.758+22.340+50.265# #:.color(purple)(=141.363cm^2# to the nearest 2 decimal places # cm^2#

Aaliyah Anderson · Answer

undefined To find the surface area of the bottom segment of the cone, we need to calculate the area of the circular base and the lateral surface area of the bottom segment.

First, calculate the area of the circular base:
Area of circular base = π * radius^2

Next, calculate the slant height of the bottom segment using the Pythagorean theorem:
Slant height = √(height^2 + (radius - cut distance)^2)

Then, calculate the lateral surface area of the bottom segment:
Lateral surface area = π * radius * slant height

Finally, add the area of the circular base and the lateral surface area to get the total surface area of the bottom segment.