A cone has a height of #16 cm# and its base has a radius of #15 cm#. If the cone is horizontally cut into two segments #4 cm# from the base, what would the surface area of the bottom segment be?

Question

A cone has a height of #16  cm# and its base has a radius of #15  cm#.  If the cone is horizontally cut into two segments #4 cm# from the base, what would the surface area of the bottom segment be?

Avery Allen · Accepted Answer

The surface area of the bottom segment is #1583.36cm^2#.

This is a truncated cone where the top segment is removed.

Let #R=15# and #h=4#,

#r=(16-4)/16xx15#
#color(white)(r)=12#

#"Surface area"=pi(r+R)(sqrt((r-R)^2+h^2))+pir^2+piR^2#
#color(white)"Surface area"=pi(12+15)(sqrt((12-15)^2+4^2))+pi(12)^2+pi(15)^2#
#color(white)"Surface area"=pi(27)(5)+pi(144)+pi(225)#
#color(white)"Surface area"=1583.36#

The surface area of the bottom segment is #1583.36cm^2#.

Claire Akin · Answer

undefined To find the surface area of the bottom segment of the cone after it is horizontally cut 4 cm from the base, we first need to find the radius of this new smaller cone formed by the bottom segment. Then, we calculate the surface area of this smaller cone.

The smaller cone's height can be found using similar triangles. Since the height of the original cone is 16 cm and the cut is 4 cm from the base, the height of the smaller cone is $16 - 4 = 12$ cm.

Using similar triangles, the ratio of the height of the smaller cone to the height of the original cone is equal to the ratio of the radius of the smaller cone to the radius of the original cone. Thus:

$$
\frac{{\text{{height of smaller cone}}}}{{\text{{height of original cone}}}} = \frac{{\text{{radius of smaller cone}}}}{{\text{{radius of original cone}}}}
$$

Substituting the known values:

$$
\frac{{12}}{{16}} = \frac{{\text{{radius of smaller cone}}}}{{15}}
$$

Solving for the radius of the smaller cone:

$$
\text{{radius of smaller cone}} = \frac{{12 \times 15}}{{16}} = 11.25 \text{ cm}
$$

Now, we find the surface area of the bottom segment of the cone using the formula for the lateral surface area of a cone:

$$
\text{{Surface area of bottom segment}} = \pi r^2
$$

Substituting the radius of the smaller cone:

$$
\text{{Surface area of bottom segment}} = \pi \times (11.25)^2
$$

$$
\text{{Surface area of bottom segment}} = \pi \times 126.5625
$$

$$
\text{{Surface area of bottom segment}} \approx 398.43 \text{ cm}^2
$$

Therefore, the surface area of the bottom segment of the cone is approximately $398.43 \, \text{cm}^2$.