A cone has a height of #18 cm# and its base has a radius of #5 cm#. If the cone is horizontally cut into two segments #12 cm# from the base, what would the surface area of the bottom segment be?

Question

A cone has a height of #18  cm# and its base has a radius of #5  cm#.  If the cone is horizontally cut into two segments #12 cm# from the base, what would the surface area of the bottom segment be?

Allison Allen · Accepted Answer

#348cm^2#

Lets first consider the cross-section of the cone.

Now it is given in the question, that AD = #18cm# and DC = #5cm#

given, DE = #12cm#

Hence, AE = #(18-12)cm = 6cm#

As, #DeltaADC # is similar to #DeltaAEF #, #(EF)/(DC) = (AE)/(AD)#

#:. EF = DC*(AE)/(AD)=(5cm)*6/18 = 5/3cm#

After cutting, the lower half looks like this:

We have calculated the smaller circle (the circular top), to have a radius of #5/3cm#.

Now lets calculate the length of the slant.

#Delta ADC# being a Right angle triangle, we can write

#AC =sqrt(AD^2+DC^2) = sqrt(18^2+5^2) cm ~~ 18.68 cm #

The surface area of the whole cone is : #pirl = pi*5*18.68 cm^2#

Using the similarity of the triangles #DeltaAEF # and #DeltaADC #, we know that all the sides of #DeltaAEF # are less than the corresponding sides of #DeltaADC # by a factor of 3.

So the slant surface area of the upper part (the smaller cone) is : #(pi*5*18.68)/(3*3)cm^2#

Hence of the slant surface area of the lower part is: #pi*5*18.68*(8/9)cm^2#

And we have the the upper and lower circular surfaces' areas as well.

So the total area is:

#pi*(5^2/3^2)_"for upper circular surface"+ pi*5*18.68*(8/9) _"for the slant surface"+ pi*(5^2) _"for lower circular surface" ~~348cm^2#

Jayden Jackson · Answer

undefined To find the surface area of the bottom segment of the cone, we first need to calculate the slant height of the cone. This can be done using the Pythagorean theorem in the right triangle formed by the height, the radius, and the slant height. The slant height (l) can be calculated as l = √(r² + h²), where r is the radius and h is the height.

Given that the radius (r) is 5 cm and the height (h) is 18 cm, we can calculate the slant height (l):

l = √(5² + 18²)
l = √(25 + 324)
l = √349
l ≈ 18.68 cm

Now, we know the slant height of the cone. Since the cone is horizontally cut into two segments 12 cm from the base, the height of the bottom segment (h') is 12 cm.

Using the formula for the lateral surface area of a cone, which is πrl, where r is the radius and l is the slant height, we can calculate the surface area of the bottom segment:

Surface area of bottom segment = π * r * l'
where r is the radius and l' is the slant height of the bottom segment.

Plugging in the values:
Surface area of bottom segment = π * 5 * 18.68
Surface area of bottom segment ≈ 294.52 square centimeters

So, the surface area of the bottom segment of the cone is approximately 294.52 square centimeters.