Volume of a truncated cone with inscribed sphere problem

[Ognjen]

Official problem formulation:

A right truncated cone is circumscribed around a sphere of radius R, with its slant height making an angle of 60 deg. with the plane of the base of the cone. Calculate the volume of the truncated cone.

The cross section of the cone is obviously a(n isosceles, since the cone is given as right in the text of the problem) trapezoid with an inscribed circle. This means that sums of the opposing sides of the trapezoid are equal to each other. Additionally, since the trapezoid is isosceles, if we mark the trapezoid with letters ABCD, side BC equals side AC. From this we conclude:

$$2r_1 + 2r_2 = AD + CB$$$$s = r_1 + r_2$$
Applying trigonometry to the triangle EBC ( with E being the point of intersection of the larger base of the trapezoid and the line perpendicular to it ), we get:

$$\sin \frac{\pi}{3} = \frac{H}{s}$$$$H = \frac{r_2 + r_1}{2} * \sqrt{3}$$
Based on the fact the trapezoid is isosceles, I can also get:

$$EB = r_1 - r_2$$
Now comes the part of the solution I do not understand where it came from.

With no justification whatsoever ( implying it is obvious most likely ), the solution puts forward the following equation:

$$BC = 2*BE$$
I tried to derive it from EB and s ( since those are essentially the two variables that define the given equation ), but didn't manage to find the right connection.

The rest of the official solution is clear from this point on ( from the H and BC equation r1 and r2 are derived ( given that H = 2R, which I find clear ), then simply plugged into the volume of a truncated cone formula ).

Can anybody provide me with an insight that would clarify the aforementioned equation and the intuition behind its conceptualization ? It would be immensely appreciated.

 

[Dr.Peterson]

The cross section of the cone is obviously a(n isosceles, since the cone is given as right in the text of the problem) trapezoid with an inscribed circle. This means that sums of the opposing sides of the trapezoid are equal to each other. Additionally, since the trapezoid is isosceles, if we mark the trapezoid with letters ABCD, side BC equals side AC. From this we conclude:

$$2r_1 + 2r_2 = AD + CB$$$$s = r_1 + r_2$$

You didn't say where A, B, C, and D are, and you surely don't mean BC=AC as you said. And I have to guess that s means the slant height. Please give us a picture so we can be sure what you are talking about.

Here is my own drawing, which may or may not match yours:

​

I think all I'd need to use is various 30-60-90 triangles. I don't think I'd use s, which is not relevant to the volume.

Now, you talk about what you did, and what "the solution" did. I can't tell how much of this is "official"; none of it looks like what I would do, though that may be because I have the wrong picture. If you're trying to understand a provided solution, please show that in its entirety before asking about it.

 

[The Highlander]

Official problem formulation:

A right truncated cone is circumscribed around a sphere of radius R, with its slant height making an angle of 60 deg. with the plane of the base of the cone. Calculate the volume of the truncated cone.

The cross section of the cone is obviously a(n isosceles, since the cone is given as right in the text of the problem) trapezoid with an inscribed circle. This means that sums of the opposing sides of the trapezoid are equal to each other. Additionally, since the trapezoid is isosceles, if we mark the trapezoid with letters ABCD, side BC equals side AC. From this we conclude:

$$2r_1 + 2r_2 = AD + CB$$$$s = r_1 + r_2$$
Applying trigonometry to the triangle EBC ( with E being the point of intersection of the larger base of the trapezoid and the line perpendicular to it ), we get:

$$\sin \frac{\pi}{3} = \frac{H}{s}$$$$H = \frac{r_2 + r_1}{2} * \sqrt{3}$$
Based on the fact the trapezoid is isosceles, I can also get:

$$EB = r_1 - r_2$$
Now comes the part of the solution I do not understand where it came from.

With no justification whatsoever ( implying it is obvious most likely ), the solution puts forward the following equation:

$$BC = 2*BE$$
I tried to derive it from EB and s ( since those are essentially the two variables that define the given equation ), but didn't manage to find the right connection.

The rest of the official solution is clear from this point on ( from the H and BC equation r1 and r2 are derived ( given that H = 2R, which I find clear ), then simply plugged into the volume of a truncated cone formula ).

Can anybody provide me with an insight that would clarify the aforementioned equation and the intuition behind its conceptualization ? It would be immensely appreciated.

It would appear that you have been given a problem accompanied by a worked solution (a part of which you are having difficulty understanding).
Please supply a complete (verbatim) copy of the original problem and a copy of the solution provided in their entirety along with a sketch/diagram of the situation with all the parameters you list clearly labelled on it, ie: $\displaystyle \text{A, B, C, D, E, H, r₁, r₂ \& s}$.
We may then be able to offer some resolution to your query.
Thank you.

 

[Ognjen]

You didn't say where A, B, C, and D are, and you surely don't mean BC=AC as you said. And I have to guess that s means the slant height. Please give us a picture so we can be sure what you are talking about.

Here is my own drawing, which may or may not match yours:

View attachment 33078​

I think all I'd need to use is various 30-60-90 triangles. I don't think I'd use s, which is not relevant to the volume.

Now, you talk about what you did, and what "the solution" did. I can't tell how much of this is "official"; none of it looks like what I would do, though that may be because I have the wrong picture. If you're trying to understand a provided solution, please show that in its entirety before asking about it.

I am very sorry, but I think I made a mistake with my reasoning.

In the official post, I said EB = r1 - r2, while it should be EB = (r1 - r2) / 2

I will provide official picture below.

My mistake, however, does not make the relation BC = 2*BE any clearer to me, unfortuantely.

 

[Dr.Peterson]

My mistake, however, does not make the relation BC = 2*BE any clearer to me, unfortuantely.

Have you never seen the 30-60-90 triangle, or observed that BCE is half of an equilateral triangle, or learned the cosine of 60 degrees? Any of those would make this immediately clear.

 

[Ognjen]

Have you never seen the 30-60-90 triangle, or observed that BCE is half of an equilateral triangle, or learned the cosine of 60 degrees? Any of those would make this immediately clear.

I'm sorry, you're right, I just noticed the relation.

However, now I have another problem, relating to the bit I corrected above.

The official solution now uses the 2 following equations to create a system of linear equations and deduce values of r1 and r2 based on R from it:

$$2R = \frac{r_1 + r_2}{2}*\sqrt{3}$$$$2(r_1 - r_2) = r_1 + r_2$$
The first equation makes sense, since H = 2R for a truncated cone, and the rest is deduced from the sin60 = H/s and the tangential quadrilateral relations.
However, the second equation doesn't, at all. While it is BASED on the equation which ( now ) makes sense to me ( 2*BE = BC ), the presumed value of BE I cannot deem correct at all.

Let's say another point M was defined as the point of interception of the line AB and a line perpendicular to the line AB from point D. Since the truncated cone is RIGHT, its cross section has to be an ISOSCELES trapezoid, meaning that lines AM and EB have to be EQUAL ( correct me if I'm wrong with any of this ). Therefore, when from the base AB ( r1 ) we subtract the base DC ( r2 ) which is normal to it ( because the cone is right ), we should get 2 equal lengths ( EB or AM ). Thus, if we were to define just one of them, we would have to divide the difference r1 - r2 by 2, right ?

If my logic were right, we would get the following equation:

$$2 * (\frac{r_1 - r_2}{2}) = r_1 + r_2$$
This simplifies to r1 - r2 = r1 + r2, which, truly, makes no sense if all sides are larger than zero ( which is always the case ). However, I don't understand why ? The foregoing logic I expounded upon seems unfalsifiable to me, where did a mistake manage to sneak up ?

 

[Dr.Peterson]

However, the second equation doesn't, at all. While it is BASED on the equation which ( now ) makes sense to me ( 2*BE = BC ), the presumed value of BE I cannot deem correct at all.

Let's say another point M was defined as the point of interception of the line AB and a line perpendicular to the line AB from point D. Since the truncated cone is RIGHT, its cross section has to be an ISOSCELES trapezoid, meaning that lines AM and EB have to be EQUAL ( correct me if I'm wrong with any of this ). Therefore, when from the base AB ( r1 ) we subtract the base DC ( r2 ) which is normal to it ( because the cone is right ), we should get 2 equal lengths ( EB or AM ). Thus, if we were to define just one of them, we would have to divide the difference r1 - r2 by 2, right ?

I'm getting very confused. The work you're doing here is to find EB = (r1 - r2) / 2, which you said was a correction. But that would only be true if r1 and r2 were diameters, not radii. And I don't see where that appears in their work.

The reasoning you show here makes sense, though I myself would just use the fact that XB=XE+EB, as seen below (where I've changed to your labeling).

​

If my logic were right, we would get the following equation:

$$2 * (\frac{r_1 - r_2}{2}) = r_1 + r_2$$
This simplifies to r1 - r2 = r1 + r2, which, truly, makes no sense if all sides are larger than zero ( which is always the case ). However, I don't understand why ? The foregoing logic I expounded upon seems unfalsifiable to me, where did a mistake manage to sneak up ?

Where did this equation come from? What lengths are you setting equal??? You need to state more clearly what you are doing at each step.

Their second equation, as I thought you recognized, comes from right triangle BCE, and specifically the fact that BC (= $r_1+r_2$) is twice EB (= $r_1-r_2$).

Actually (I'm gradually figuring out what you're talking about) both equations come from this same triangle; the first is from the sine of 60, and the second from the cosine, in effect.