The maggot and the apple.

Pole Monkey

A maggot eats through the middle of a spherical apple leaving a cylindrical hole of length L.

How would you calculate the remaining volume of the apple?

Victor

How accurate does the answer need to be?

Mathematical answer: It is approximately a sphere minus a cylinder.

Physics answer: use the displacement method.

Pole Monkey

Gotta be exact, this is the maths forum.

cruizer101

In that case you have the sphere minus a cylinder of length L, where is L is shorter than the diameter by 2h, h being the height of the spherical cap at end of cylinder, then also take the volumes of the spherical caps away from sphere too.

Pole Monkey

Do the math, do the math!

10-10-20

Victor wrote: »

Mathematical answer: It is approximately a sphere minus a cylinder.

...plus the two spherical caps where the cylinder meets the outside surface of the sphere.
And as we don't know the diameter of the maggot at the point of entry, nor whether he gained weight during his trip through the apple, nor whether his intended direct path was modified when he hit a pip making it less than a direct route from A to B, nor whether the apple suffered a loss of fluids through his penetration causing slight shrinkage... did he defecate?

Pole Monkey

No-one going to do the math?

I've done it and it's quite surprising...

Yakuza

As best as I can see (and borrowing a formula from wikipedia), assuming r is the radius of our spherical apple, the remaining volume of the apple minus a cylinder of length L and radius sqrt(r²-L²/4) and two spherical caps of height (r-L/2) (and radius r) should be:
4/3*pi*r³ - pi*(r²-L²/4)*L - 2 * (pi/3 *(r-L/2)²*(2r-L/2)). It doesn't exactly trip off the tongue.

That's a rough first attempt at any rate, I haven't bothered to expand it out or gather terms.

Pole Monkey

Look at it this way. The radius of the hole, r, and the sphere, R, aren't independent. They are related by the equation,

[latex]R^2 = r^2 + L^2/4[/latex]

so we can express the volume of the cylinder as,

[latex]V_{cyl} = {\pi}LR^2 - {\pi}L^3/4[/latex]

and the volume of the caps as,

[latex]V_{caps}=4{\pi}R^3/3-{\pi}R^2L+{\pi}L^3/12 [/latex]

The remaining volume is then,

[latex]V_{sphere }-V_{cyl} - V_{caps}={\pi}L^3/6[/latex]

What's happened to R? It's canceled out! The remaining volume only depends on the length of the hole, not its radius or the radius of the apple. A giant space maggot could eat through an apple the size of the Earth and leave behind the same amount of apple as a baby maggot eating through a baby apple as long as the lengths of the holes are the same.

WTF?

13th Skull

Dat's rubbish. How could a little baby maggot eat as much as a massive one?

Eliot Rosewater

Pole Monkey wrote: »

What's happened to R? It's canceled out!

It hasn't; it's merely concealed. As you said yourself: The radius of the hole, r, and the sphere, R, aren't independent. They are related by the equation,

[latex]R^2 = r^2 + L^2/4[/latex]

This gives

[latex] L=2 \sqrt{ R^2 - r^2} [/latex]

and that the remaining volume is

[latex]\displaystyle \frac{\pi}{6} 2^3 \left( R^2 - r^2 \right)^{3/2} [/latex]

This seems to make sense: bigger apple (bigger R) = bigger volume left over; bigger maggot (bigger r) = smaller volume left over.

Mellor

Eliot Rosewater wrote: »

It hasn't; it's merely concealed. As you said yourself: The radius of the hole, r, and the sphere, R, aren't independent. They are related by the equation,

[latex]R^2 = r^2 + L^2/4[/latex]

Maybe I'm just slow because I'm hungover, but how are they not independent?
Could the same maggot not eat through two different sized apples leaving the same diameter holes? Or a big and small maggot each eating the same sized apples leaving two diffetent holes etc

Eliot Rosewater

Mellor wrote: »

Maybe I'm just slow because I'm hungover, but how are they not independent?
Could the same maggot not eat through two different sized apples leaving the same diameter holes? Or a big and small maggot each eating the same sized apples leaving two diffetent holes etc

Yeah, you're right! The statement should say "the quantities R, r and L are not independent," where L is the length of the hole minus the caps. Then the answer depending on L does not mean it doesn't depend on R and r.

13th Skull

Eliot Rosewater wrote: »

Then the answer depending on L does not mean it doesn't depend on R and r.

No, it does only depend on L. Given a length, L, we can determine the remaining volume regardless of R and r. That's the beauty.

bnt

I suspect this is a Calculus question - something to do with volumes of solids of revolution, perhaps? That's the first thing I thought of, but I wasn't very good at it ...

sock puppet

Or you could just skip all the maths and use simulation to work it out...

TheBody

sock puppet wrote: »

Or you could just skip all the maths and use simulation to work it out...

Be gone from this forum!! Where are the mods when you need them!! :pac:

MathsManiac

TheBody wrote: »

Be gone from this forum!! Where are the mods when you need them!! :pac:

Looking on in wry amusement...