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#1 2009-05-16 20:30:10

kuku
Member
Registered: 2009-05-16
Posts: 2

Applications of numerical integrals to practical volumes

Hi everyone!

can anyone help me with those 2 maps questions

1_ A very large standing stone was erected by an early civilisation as a sentinal. Archaeologists measured the girth (circumference) of the stone at heights seprated by 1m. They found the girths to be 6.3m (at ground level), 5.4m, 4.6m, 4.4m, 4.8m, and 3.9m, and then the stone tapered to a point at a height of 6m. Assuming that the stone is made of basalt of density 2800kg/m^3 and is nearly square in cross-section, find an approximation for the mass of the stone.

2_ The circumference of the trunk of a stringybark is measured at ground height and at heights separated by 5m, with the following results: 8.4m, (ground), 8.3m, 8.2m, 7.4m, 6.3m, 54,m 4.8m, 3.6m, 3.4m, 3.2m, 2.7m, 2.1m, 1.4m, 1.2m, 0.9m, 0.6m, 0.4m, 0.2m, 0.1m, then tapering to the top at a height of 95m. Find an approximation for the volume of wood in the log from this tree.

If anyone can draw the diagrams plz do it.
thanks

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#2 2009-05-17 03:47:28

G_Einstein
Member
Registered: 2008-08-30
Posts: 124

Re: Applications of numerical integrals to practical volumes

Try to solve this may be by double or triple integral.


Se Zoti vete e tha me goje,se kombet shuhen permbi dhe,por SHqiperia do te roje,per te,per te luftojme ne.
God said that all nation exincts on the ground,but Albania will survive,for it,for it we are fighting.

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#3 2009-05-17 05:58:12

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Applications of numerical integrals to practical volumes

A single integral will work, you just need to find the function that relates height to cross-sectional area.

Probably the best approximation for girth is to use the trapezium rule.
ie. At h = 0, g = 6.3
    At h = 1, g = 5.4
    At h = x, g = 6.3 - 0.9x (for 0<x<1)

etc.

You're told that each cross-section is a square, and so you can conclude that its side length is g/4 and so its area is g²/16.

The volume of the stone is therefore 1/16∫(g(x))²dx, where g(x) is the girth function.

The second question is similar to the first, but this time each cross-section is a circle, so the area will relate to the circumference differently.


Why did the vector cross the road?
It wanted to be normal.

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#4 2009-05-17 07:56:29

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Applications of numerical integrals to practical volumes

Hi mathsyperson;

  The problem 1 speaks of a circumference making me think this is a large cone, but then says its cross section (I am assuming he means the base) is nearly square, reminding me of a pyramid. Which is right, what is the shape of this stone?

Last edited by bobbym (2009-05-17 07:58:33)


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2009-05-17 08:42:37

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Applications of numerical integrals to practical volumes

The phrasing of the question makes me think that it's closer to a pyramid than a cone, but I admit it's ambiguous.

I also forgot to mention that it's the mass that's asked for, so after the volume is found you need to combine it with the density to get the answer.


Why did the vector cross the road?
It wanted to be normal.

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#6 2009-06-04 00:55:46

kuku
Member
Registered: 2009-05-16
Posts: 2

Re: Applications of numerical integrals to practical volumes

thx everyone 4 help

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#7 2009-06-04 11:47:31

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Applications of numerical integrals to practical volumes

Hi kuku;

Can you shed some light on what the true shape of the object is. Mathsyperson and I believe it is pyramidal.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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