A cone has a height of #12 cm# and its base has a radius of #6 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 #12  cm# and its base has a radius of #6  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?

Sadie Allen · Accepted Answer

Total surface area of bottom segment is #331.04(2dp) cm^2 # The cone is cut at 6 cm from base, So upper radius of the frustum of cone is #r_2=(12-6)/12*6=3 cm # ; slant ht #l=sqrt(6^2+(6-3)^2)=sqrt(36+9)=sqrt 45=6.71(2dp) cm# Top surface area #A_t=pi*3^2=28.27 cm^2# Bottom surface area #A_b=pi*6^2=113.10 cm^2# Slant Area #A_s=pi*l*(r_1+r_2)=pi*6.71*(6+3)=189.67 cm^2# Total surface area of bottom segment #=A_t+A_b+A_s=28.27+113.10+189.67=331.04(2dp) cm^2 #[Ans]

Caroline Aucoin · Answer

undefined To find the surface area of the bottom segment of the cone, we need to calculate the curved surface area of the cone and subtract the area of the smaller cone created by the horizontal cut.

The formula for the curved surface area of a cone is: π × radius × slant height.

First, we need to find the slant height (l) of the original cone using the Pythagorean theorem:
l = √(h^2 + r^2), where h is the height and r is the radius of the cone.

Substituting the given values:
l = √(12^2 + 6^2)
l = √(144 + 36)
l = √180
l ≈ 13.42 cm (rounded to two decimal places).

Now, we can find the curved surface area (CSA) of the original cone:
CSA = π × r × l
CSA = π × 6 × 13.42
CSA ≈ 253.55 cm² (rounded to two decimal places).

Next, we calculate the radius (r') and slant height (l') of the smaller cone created by the horizontal cut:
r' = r = 6 cm (radius remains the same)
h' = h - 6 = 12 - 6 = 6 cm (height is reduced by 6 cm)
l' = √(h'^2 + r'^2)
l' = √(6^2 + 6^2)
l' = √(36 + 36)
l' = √72
l' ≈ 8.49 cm (rounded to two decimal places).

Now, we can find the curved surface area (CSA') of the smaller cone:
CSA' = π × r' × l'
CSA' = π × 6 × 8.49
CSA' ≈ 160.92 cm² (rounded to two decimal places).

Finally, to find the surface area of the bottom segment, we subtract CSA' from CSA:
Surface area of bottom segment = CSA - CSA'
Surface area of bottom segment ≈ 253.55 cm² - 160.92 cm²
Surface area of bottom segment ≈ 92.63 cm² (rounded to two decimal places).

Therefore, the surface area of the bottom segment of the cone is approximately 92.63 square centimeters.