Geometry problem

[Zero_exe]

I really need help on problem 06H-50. Ive tried plugging in numbers but I keep getting different values.

[AuSmith]

There are multiple ways to do this one.

1) I would use the fact that the two cones are similar and the frustum is merely the difference of the two cones. The volume of the smaller cone is [unparseable or potentially dangerous latex formula] for some constant of proportionality [unparseable or potentially dangerous latex formula], and the volume of the larger cone is [unparseable or potentially dangerous latex formula]. Since the volume of the frustum is equal to that of the smaller cone, the larger cone has twice as much volume as the smaller.

[unparseable or potentially dangerous latex formula]

Reduce,

[unparseable or potentially dangerous latex formula]

Notice that [unparseable or potentially dangerous latex formula] is irrelevant to the problem. You can set it to [unparseable or potentially dangerous latex formula].

2) You could set the apex angle to [unparseable or potentially dangerous latex formula] as well as [unparseable or potentially dangerous latex formula]. If they don't give you information and you can still meet the hypothesis of the problem, then choose all the numbers you want. Then, you could use the [unparseable or potentially dangerous latex formula] to find the volumes of the two cones. Again, the bigger should be twice the smaller.

[mathslug]

I saw this question and had to attempt it. I came up with an answer, but its not as elegant as Aaron's. Here's what I did:

Let R1 be the radius of the top of the frustrum, R2 be the base radius, and h be the height as indicated in the diagram.

Lets make use of the fact that the volume of the frustrum is equal to the volume of the cone:

(1/3)(pi)(h){(R1^2) + (R1)(R2) +(R2^2)} = (1/3)(pi)(R1^2)(x)(h) -->

x = {(R1^2) + (R1)(R2) +(R2^2)} / (R1^2)

It seems like a dead end, right? WRONG. The small cone and big cone can be seen as similar triangles, which allows us to make a relationship between the two:

(xh) / (xh + h) = R1 / R2 -->

R2 = R1 * (x+1)/x

plugging back in and reducing yields the following result:

0 = [(x+1)/x]^2 + [(x+1)/x] + 1 - x

I plugged this into the solver and got that x = 3.85

Technically this is a 3 step problem. 1) solve for x in terms of R1 and R2 using frustrum and cone volume formulas. 2) Create a relationship between R1 and R2 using the fact that the big and small cones are similar triangles, and 3) Plug this relationship into the formula you obtained from 1), and then use the solver to solver for x.

Yay!

Zack