A cone has a height of #6 cm# and its base has a radius of #2 cm#. If the cone is horizontally cut into two segments #5 cm# from the base, what would the surface area of the bottom segment be?

Question

A cone has a height of #6  cm# and its base has a radius of #2  cm#.  If the cone is horizontally cut into two segments #5 cm# from the base, what would the surface area of the bottom segment be?

Autumn Armstrong · Accepted Answer

Total surface area of bottom segment is #51.55# sq.cm.

The cone is cut at 5 cm high from base, So upper radius of the frustum of cone is #r_2=(6-5)/6*2=1/3 =0.33#cm ; Slant height of the frustum of cone is #l=sqrt(5^2+(2-1/3)^2)=sqrt(25+2.78)~~5.27# cm Top surface area #A_t=pi*0.33^2 ~~0.35 # sq.cm Bottom surface area #A_b=pi*2^2=12.57 # sq.cm Slant Area #A_s=pi*l*(r_1+r_2)=pi*5.27*(2+0.33)~~38.63# sq.cm Total surface area of bottom segment is #=A_t+A_b+A_s=0.35+12.57+38.63 ~~51.55 (2dp)# sq.cm [Ans]

Carson Arnold · Answer

undefined To find the surface area of the bottom segment of the cone, we first need to find the slant height of the cone. Then, we can calculate the lateral surface area of the bottom segment using the formula for the lateral surface area of a cone segment.

The slant height (l) of the cone can be found using the Pythagorean theorem, where l is the hypotenuse, r is the radius of the base, and h is the height of the cone:

l = √(r^2 + h^2)

Substituting the given values:

l = √(2^2 + 6^2)
  = √(4 + 36)
  = √40
  = 2√10

Now, we can calculate the lateral surface area (A) of the bottom segment of the cone using the formula:

A = πr(l - r)

Substituting the given values:

A = π * 2 * (2√10 - 2)
  = 2π(√10 - 1)

Therefore, the surface area of the bottom segment of the cone is 2π(√10 - 1) square centimeters.