A cone has a height of #8 cm# and its base has a radius of #5 cm#. If the cone is horizontally cut into two segments #6 cm# from the base, what would the surface area of the bottom segment be?

Question

A cone has a height of #8  cm# and its base has a radius of #5  cm#.  If the cone is horizontally cut into two segments #6 cm# from the base, what would the surface area of the bottom segment be?

Sophia Alfred · Accepted Answer

Total surface area of bottom segment is #222.38# sq.cm

The cone is cut at 6 cm from base, So upper radius of the frustum of cone is #r_2=(8-6)/8*5=1.25# cm ; Slant ht #l=sqrt(6^2+(5-1.25)^2)=sqrt(36+14.0625)# #=sqrt 50.0625 ~~ 7.08# cm Top surface area #A_t=pi*1.25^2 ~~ 4.91#sq.cm Bottom surface area #A_b=pi*5^2 ~~ 78.54# sq.cm Slant Area #A_s=pi*l*(r_1+r_2)=pi*7.08*(5+1.25)# #~~138.93# sq.cm Total surface area of bottom segment #=A_t+A_b+A_s=4.91+78.54+138.93# # ~~ 222.38# sq.cm [Ans]

Sadie Allen · Answer

undefined To find the surface area of the bottom segment of the cone, we first need to calculate the slant height of the bottom segment using the Pythagorean theorem.

The slant height (l) can be found using the formula:

l = √(r^2 + h^2)

Where:
r = radius of the base = 5 cm
h = height of the cone = 8 cm

Plugging in the values:

l = √(5^2 + 8^2) = √(25 + 64) = √89 ≈ 9.43 cm

Now, since the segment is 6 cm from the base, the height of the bottom segment is 8 cm - 6 cm = 2 cm.

The surface area of the bottom segment can be calculated using the formula for the lateral surface area of a cone segment:

A = πrl

Where:
A = surface area of the segment
r = radius of the base = 5 cm
l = slant height of the segment ≈ 9.43 cm

Plugging in the values:

A = π * 5 * 9.43 ≈ 47.15 cm²

Therefore, the surface area of the bottom segment of the cone is approximately 47.15 square centimeters.