Can't Dig to China: Man Finds Out The Hard Way

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In summary, a man drills a cylindrical tunnel through a rosewood globe to test the theory of digging a hole straight down from America to China. However, he finds that the other end of the tunnel is in the southern hemisphere. The question is asked about the volume of material left in the globe after drilling the tunnel, which is dependent on the length of the tunnel measured along its wall and can be calculated using the equations provided. The radius of the tunnel is not necessary to find the volume.
  • #1
Gokul43201
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I hope this is not old too...

A man decides the find out for himself whether or not "if you dig a hole straight down from America, you get to China". So he grabs his beautiful rosewood globe and drills a cylindrical tunnel diametrically through it, starting from America. He finds the other end is far from China... in fact it's in the southern hemispere.

So here's the question : If the length of the tunnel measured along its wall is 'L', what the volume of the material left in the globe after drilling out the tunnel ?
 
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  • #2
Gokul43201 said:
I hope this is not old too...

A man decides the find out for himself whether or not "if you dig a hole straight down from America, you get to China". So he grabs his beautiful rosewood globe and drills a cylindrical tunnel diametrically through it, starting from America. He finds the other end is far from China... in fact it's in the southern hemispere.

So here's the question : If the length of the tunnel measured along its wall is 'L', what the volume of the material left in the globe after drilling out the tunnel ?

Surely the radius of the tunnel is important too?

As r tends to zero the volume tends to:

[tex]\pi(\frac{L^3}{6} - Lr^2)[/tex]

where r is the radius of the tunnel.
 
  • #3
jcsd said:
Surely the radius of the tunnel is important too?

No, the question is complete !
 
  • #4
Of course! The length of the tunnel is dependent on it's radius, so it's going to disappear in the final equation.
 
  • #5
Okay the missing parts of the globe are made up of the volume of the tunnel plus extra missing volume from the ends due to curved surface of the globe.

If we define R as the radius of the globe and r as a radius of a section of the circle of the tunnel we can define a length a:

[tex]R^2 - a^2 = r^2[/tex]

As when a = L/2, r = the radius of the tunnel, the volume of the tunnel is given by:

[tex]\pi LR^2 - \frac{\pi L^3}{4}[/tex]


To find the area of the 'missing volume' at the ends we can us tehe fact that a radius of a cross section is

[tex]\pi R^2 - \pi a^2[/tex]

So the volume of the 'missing ends' is given by (the factor of 2 appears as there are 2 of them):
[tex]2\int^{R}_{L/2} (\pi R^2 - \pi a^2) da = 2\left[\pi R^2 a - \frac{\pi a^3}{3}\right]^{R}_{L/2}[/tex]
[tex]= \frac{4R^3\pi}{3} - \pi L R^2+ \frac{\pi L^3}{12}[/tex]

therfore the volume of left is (the volume of the sphere - the volume of the cynclinder and the 'ends':

[tex]\frac{4R^3\pi}{3} - \frac{4R^3\pi}{3} + L\pi R^2 - \frac{\pi L^3}{12} - \pi LR^2 + \frac{\pi L^3}{4}[/tex]

[tex]=\frac{\pi L^3}{6}[/tex]

(barring any silly mistakes).
 
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  • #6
Neat problem. My initial reaction also was that we would need the radius of the tunnel. Tricky.



Njorl
 
  • #7
Infact I've just relaized looking at my first post, if you know that the voulem is dependt only on L you can guess from the fact that as r tends to zero the volume tends to:

[tex]\frac{\pi L^3}{6}[/tex]

without haveing to integrate.
 
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  • #8
jcsd said:
Infact I've just relaized looking at my first post, if you know that the voulem is dependt only on L you can guess from the fact that as r tends to zero the volume tends to:

[tex]\frac{\pi L^3}{6}[/tex]

without haveing to integrate.

Exactly !
jcsd with the point (I'm feeling generous, so I'll ignore the spellings).
 
  • #9
Gokul43201 said:
Exactly !
jcsd with the point (I'm feeling generous, so I'll ignore the spellings).

oi! there called typos :biggrin:
 
  • #10
And don't say: they're called typos.
 
  • #11
Okay ! Guess it's hard to type correctly when you're drowning.
 

1. What does it mean to "dig to China"?

"Digging to China" is a popular phrase used to describe the act of digging a hole straight through the center of the Earth. This is not actually possible due to the Earth's molten core and the laws of physics.

2. Why did the man in the story try to dig to China?

The man in the story most likely attempted to dig to China out of curiosity or as a fun experiment. He may have also believed in the myth that if you dig deep enough, you will end up in China.

3. What happened to the man when he tried to dig to China?

The man likely encountered many obstacles such as rocks, soil, and possibly water as he dug deeper into the ground. Eventually, he would have reached a depth where the heat and pressure from the Earth's core would have made it impossible to continue digging any further.

4. Is it possible to dig to China from any location on Earth?

No, it is not possible to dig to China from any location on Earth. The location of China in relation to other countries and bodies of water would not allow for a straight path through the Earth's center.

5. What is the deepest hole that has ever been dug by humans?

The deepest hole ever dug by humans is the Kola Superdeep Borehole in Russia, which reached a depth of 7.5 miles (12 kilometers). This is still a far cry from the 8,000-mile distance to the center of the Earth.

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