Find an expression for the height of this frustum

[Natasha1]

Homework Statement

A frustum is made by removing a small cone from a large cone. The cones are mathematically similar.

(please see picture attached)

The large cone has base radius r cm and height hcm. Given that

volume of frustum/volume of large cone = 98/125

Find an expression, in terms of h, for the height of the frustum.

The Attempt at a Solution

By rearranging volume of frustum/volume of large cone = 98/125

we get:

volume of frustum = 1/3 x pi x r^2 x h x 98/125

Is that the right start? Do I need to work the volume of the frustum?

 

[Natasha1]

Anyone?

 

[kuruman]

You need to work out an expression for the ratio of the volumes first in terms of radii and heights. Then note that there are similar triangles to exploit.

 

[Natasha1]

Volume of large cone is 1/3.pi.r.h = 125
Volume of small cone is 1/3.pi.R.H = 27

note:
radius r of large cone is different to radius R of small cone
height h of large cone is different to height H of small cone

Where do I need to go from here?

 

[kuruman]

Volume of large cone is 1/3.pi.r.h = 125
Volume of small cone is 1/3.pi.R.h = 27

radius r of large cone is different to radius R of small cone

Where do I need to go from here?

The ratio of the cone volumes is $\frac{27}{125}$ but this does not mean that the small cone is 27 and the large cone is 125. If the ratio of my age to my father's age is $\frac{1}{2}$, this doesn't mean that I am 1 year old and my father is 2 .

Write the ratio $r$ of the volumes in terms of symbols first. Be sure to use the correct formula for the volume of a cone and use different symbols for different quantities. Also, it would help tremendously if you used LaTeX to write your formulas and equations. Click on the LaTeX link on the lower left, next to the question mark. It's easy to learn and will serve you well.

 

[Natasha1]

I can't do latex sorry...

1/3.pi.R.H / 1/3.pi.r.h leaving you with R.H / r.h

 

[kuruman]

I can't do latex sorry...

1/3.pi.R.H / 1/3.pi.r.h leaving you with R.H / r.h

The way you type in your equations is confusing and can be easily misinterpreted which makes it unacceptable. If you can't do LaTeX, then you should post photographs of your equations. Make sure the writing is legible, use a black pen for enhanced contrast and plenty of light.

The expression you show is incorrect. You will need to look up the formula for the volume of a cone. Also, simplify your expression as much as you can.

 

[Natasha1]

Please see my work

 

[kuruman]

That's very legible thank you, but it's incorrect. It seems you did not look up the formula for the volume of a cone. Please look it up and correct your work.

 

[Natasha1]

Oh, yes there should be R^2 and r^2. But I'm still stuck... Could you please start me off and then I can probably finish it...

 

[kuruman]

Please post your corrected expression for the ratio so that I can refer to it.

 

[Natasha1]

Here it is

 

[kuruman]

Very good. Now make a drawing of the two cones. Label all four quantities that appear in your expression for the ratio. Can you find an expression of the ratio of the radii in terms of the ratio of the heights?

 

[Natasha1]

Am I there yet? What is left for me to do?

 

[kuruman]

Has this uploaded?

It uploaded but you took a wrong turn. You have the expression
$$\frac{V_{small~cone}}{V_{large~cone}}=\frac{R^2H}{r^2h}.$$You want to use the similar triangles to eliminate the radii, not the heights.

 

[Natasha1]

I give up now... this is y final work but not sure what I have done really and where I have gone wrong

 

[kuruman]

Don't give up so quickly because you are almost there . You did not quite understand what I said in #13. Look at the equation$$\frac{V_{small~cone}}{V_{large~cone}}=\frac{R^2H}{r^2h}=\left(\frac{R}{r}\right)^2\frac{H}{h}.$$You have already found that $$\frac{R}{r}=\frac{H}{h}.$$What do you get when you use the second equation to eliminate the ratio of the radii in the first equation?

 

[Natasha1]

How do I go from what I wrote (see picture) to find an expression, in terms of h, for the height of the frustum?

 

[kuruman]

How do I go from what I wrote (see picture) to find an expression, in terms of h, for the height of the frustum?

So far
(a) In post #4 you found that the ratio of the volumes of the small cone to the large cone is $\frac{27}{125}$.
(b) You just found that this ratio is also $\frac{H^3}{h^3}.$

What do you get when you put these two together? How is the height of the frustum related to $H$ and $h$?