find an expression, in terms of h, for the height of the frustrum

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bumblebee123

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I've been puzzling over a maths past paper question for a long time, but I still can't figure it out. can anyone explain the answer to me?

Question: a frustum is made by removing a small cone from a large cone. the cones are mathematically similar

the cone has base radius r cm and height h cm

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

find an expression, in terms of h, for the height of the frustum

I know this must be to do with scale factors, but whatever I try makes no sense and gives me the wrong answer. any help would be really appreciated :)
 
I would begin by writing:

[MATH]\frac{98}{125}=1-\frac{27}{125}=1-\left(\frac{3}{5}\right)^3[/MATH]
This means the height of the smaller cone removed from the larger to form the frustum is 3/5 as large as the the height of the large cone, hence the height of the frustum must be:

[MATH]h_F=\frac{2}{5}h[/MATH]
 
MarkFL said:
I would begin by writing:

[MATH]\frac{98}{125}=1-\frac{27}{125}=1-\left(\frac{3}{5}\right)^3[/MATH]
This means the height of the smaller cone removed from the larger to form the frustum is 3/5 as large as the the height of the large cone, hence the height of the frustum must be:

[MATH]h_F=\frac{2}{5}h[/MATH]
Click to expand...


I don't really understand the first part. how did 1 - (3/5)^3 turn into just 3/5 of the height of the large cone?
 
The volume of any 3-dimensional solid is proportional to the cube of any of its linear measures. For similar shapes, the constant of proportionality is the same. Does that help?
 
bumblebee123 said:
I've been puzzling over a maths past paper question for a long time, but I still can't figure it out. can anyone explain the answer to me?

Question: a frustum is made by removing a small cone from a large cone. the cones are mathematically similar

the cone has base radius r cm and height h cm

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

find an expression, in terms of h, for the height of the frustum

I know this must be to do with scale factors, but whatever I try makes no sense and gives me the wrong answer. any help would be really appreciated :)
Click to expand...


GOOD AFTERNOON, congratulations are excellent mathematics and I have the full assurance that I will learn a lot from you and I hope that you of me, I am ready for any query and I hope we continue in continuous communication.
 
MarkFL said:
The volume of any 3-dimensional solid is proportional to the cube of any of its linear measures. For similar shapes, the constant of proportionality is the same. Does that help?
Click to expand...

ah okay! So does this mean that the surface area would be proportional to the square of any of its linear measures? does it change because the volume is measured in cm^3 and the surface area is measured in cm^2
 
bumblebee123 said:
ah okay! So does this mean that the surface area would be proportional to the square of any of its linear measures? does it change because the volume is measured in cm^3 and the surface area is measured in cm^2
Click to expand...

Yes, the surface area would be proportional to the square of any linear measure. It changes because of the number of dimensions that are involved in areas vs. volumes. :)
 
bumblebee123 said:
I'm still not too sure about why it would be 1- ( 3/5 )^3
Click to expand...

That is equivalent to the given ratio of 98/125. By expressing this ratio in the form \(1-a^3\), we know the height of the smaller cone removed relative to the height of the larger cone from which the smaller was removed.
 
bumblebee123 said:
Question: a frustum is made by removing a small cone from a large cone. the cones are mathematically similar

the cone has base radius r cm and height h cm

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

find an expression, in terms of h, for the height of the frustum
Click to expand...
What I would do is to start by noting that if the frustum (the piece left when the smaller cone is removed) is 98/125 of the larger cone, then the piece removed is 1 - 98/125 = 27/125 of the larger. Since this is (3/5)^3, the linear scale factor is 3/5.
 
ADRIANA DELGADO said:
GOOD AFTERNOON, congratulations are excellent mathematics and I have the full assurance that I will learn a lot from you and I hope that you of me, I am ready for any query and I hope we continue in continuous communication.
Click to expand...
Sooooo....what will you be trying to sell to us, Adriana?
 
after looking at what everyone has said for a very long time, I think I finally understand it:

small cone = 1 - frustum = 1 - ( 98/125 ) = 27/125

the scale factor of small cone = cube root of 27/125 because the volume would be proportional to the cube of its linear measure

= 3/5

scale factor of frustum = 1 - scale factor of small cone = 1 - 3/5 = 2/5

frustum = 2/5 h
 
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