A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If a particular frustum has an altitude of 24 centimeters, the area of its lower base is 225\pi sq cm and the area of its upper base is 25\pi sq cm, what is the altitude of the small cone that was cut off?

Guest Jun 19, 2018

#1**+2 **

area of lower base = 225π cm^{2}

and...

area of lower base = π(radius of lower base)^{2}

So...

π(radius of lower base)^{2} = 225π cm^{2}

Divide both sides of the equation by π .

(radius of lower base)^{2} = 225 cm^{2}

Take the positive square root of both sides.

radius of lower base = 15 cm

area of upper base = 25π cm^{2}

and...

area of upper base = π(radius of upper base)^{2}

So...

π(radius of upper base)^{2} = 25π cm^{2}

Divide both sides by π .

(radius of upper base)^{2} = 25 cm^{2}

Take the positive square root of both sides.

radius of upper base = 5 cm

So this is what a cross section of the cone perpendicular to the base looks like:

where h is the altitude of the small cone that was cut off, and each length is in cm.

The bases of the frustrum are parallel, and corresponding angles are congruent,

So by AA similarity, we can say that the large right triangle and the small right triangle are similar.

So...

\(\frac{\text{height of large triangle}}{\text{height of small triangle}}=\frac{\text{base of large triangle}}{\text{base of small triangle}}\\~\\ \frac{24+h}{h}=\frac{15}{5}\\~\\ \frac{24+h}{h}=3\\~\\ 24+h=3h\\~\\ 24=2h\\~\\ 12=h\)

.hectictar Jun 19, 2018

#2**0 **

Great work hectictar. I got the same answer but you were just ahead of me!.

Guest Jun 19, 2018

edited by
Guest
Jun 19, 2018

#3**+2 **

Yes, Hectictar, great work (as usual). Because of your well-presented solution, and your speed in posting it, you’ve saved the forum from having to bear a yet another pile of chaotic and indeterminate SLOP in the form of Fake Math.

**We thank you! **

GingerAle
Jun 19, 2018

#4**0 **

The guest didn't post his answer, what makes you think that his solution to the question is sloppy?

Guest Jun 19, 2018

#5**+2 **

For the same reason that when I toss a banana peel, it falls toward the Earth whether I see it, or not.

Anyway, the guest did post an answer and then removed it.

GingerAle
Jun 19, 2018

#7**+1 **

**Yes, I saw it. I’m not going to reproduce it because I don’t have enough Dramamine.**

**However, here’s a paraphrase of the FAKE MATH highlights.**

You wrote:* sqrt(225) = 25, so the radius of the lower base is 25.*

(The last time I checked 15 was the sqrt of 225)

You wrote,* the height of the big triangle is 24 *

(It’s actually 24 + h)

You used this to “solve” it

h/24 = 5/25

h = 4.8

The only thing interesting about your post is the coincidental "correct" result.

You are user TPM. This is a DNA match for the crap he'd post!

**You used FAKE MATH, Blarney, and BS, and you just dumb-lucked into Hectictar’s correct answer, in time to replace your BS with more BS.**

GA

GingerAle
Jun 20, 2018