A cone has a height of #8 cm# and its base has a radius of #6 cm#. If the cone is horizontally cut into two segments #7 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 #6  cm#.  If the cone is horizontally cut into two segments #7 cm# from the base, what would the surface area of the bottom segment be?

Aaliyah Anderson · Accepted Answer

Area is #765/8pi# #cm^2~~300.42# #cm^2#

We can see that the area of the frustum is just the difference between the area difference of the sector 2 concentric circles.
This area difference is essentially the formula for the area of a frustum:
#A=pi(R+r)sqrt((R-r)^2+h^2)#

Ignoring Figure 2 and its labels, #R# here is the big circle's radius and #r# is the smaller circle's radius, and #h# is the height of the frustum.

Back to Figure 1, the height #h# of the bottom portion of the cone is #7cm# as stated in the problem, and the radius #R# of the bottom is #6cm#. Now we just need to find #r#.

Recall #tantheta=(opposite)/(adjacent)#

In the cone, image #2theta# as the angle at the vertex of the cone (refer to Figure 1). In that case, the opposite side would be the radius of the cone, #R# and the adjacent side would be the height of the cone, #h#.

#tantheta=R/h=6/8=3/4#

However, the opposite side of #theta# can also be #r#, the radius of the top of the frustum and the adjacent side can also be #h-7cm# or #1#. In this case:

#tantheta=r/(h-7)=r#

Substitute #tantheta=3/4#:
#r=3/4#

Phew!

Now, we can finally calculate the area of the frustum! #A=pi(R+r)sqrt((R-r)^2+h^2)#
where #R=6cm#, #h=7cm#, #r=3/4cm#
#A=(945)/(16)pi# #cm^2#

Now we can calculate the areas of the top and bottom circles easily and add it all together:

#A_("total")=945/16pi+9/16pi+36pi#
#=pi(945/16+9/16+36)#
#=765/8pi# #cm^2#
#~~300.42# #cm^2#

#:.# The surface area of the bottom portion is around #300.42# #cm^2#

Jayden Jackson · Answer

undefined To find the surface area of the bottom segment of the cone:

1. Calculate the slant height ($l$) of the cone using the height ($h$) and the radius ($r$) of the base using the Pythagorean theorem:
$$ l = \sqrt{r^2 + h^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \, \text{cm} $$

2. Determine the radius ($r_1$) of the circular base of the bottom segment. This can be found using similar triangles:
$$ \frac{r_1}{r} = \frac{h_1}{h} $$
$$ \frac{r_1}{6} = \frac{1}{8} \times (8 - 7) $$
$$ r_1 = \frac{1}{8} \times 1 \times 6 = \frac{3}{4} \, \text{cm} $$

3. Calculate the surface area ($A_1$) of the bottom segment of the cone using the formula for the surface area of a cone:
$$ A_1 = \pi r_1 l = \pi \times \frac{3}{4} \times 10 = \frac{30\pi}{2} = 15\pi \, \text{cm}^2 $$

So, the surface area of the bottom segment of the cone is $ 15\pi \, \text{cm}^2 $.