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### Speed of Peeling Potatoes with Different Surface Areas

```Date: 09/11/2006 at 09:34:49
From: Drew
Subject: Physics speed comparison with varying surface areas and mass

Joe and Harry have been assigned the task of peeling potatoes.  Joe is
given 40 kg of potatoes, which average 1 kg in size, while Harry is
given 20 kg of potatoes, which average 0.5 kg in size.  Assuming that
Joe's and Harry's peeling skills are equal, and if Joe finishes his
task in one hour, how long will it take Harry to accomplish his task
and why?

I'm having trouble understanding the answer given, and where I went
wrong in my answer.  This is from the first unit test in my physics
course.  My teacher's explanation is given below and I cannot
understand what it means in comparison to what I was thinking.  Please
help.  Thanks!

Teacher's Answer: Approximately 48 minutes because the time required
to peel a potato is proportional to the surface area of the potato,
while the mass is proportional to its volume, so the ratio of the
times is proportional to the 2/3 power of the ratio of the masses.

My thinking:

SA of a sphere = 4(3.14)r^2
V of a sphere  = 4/3(3.14)r^3

V(Joe): 1 = 4/3(3.14)r^3, r = .62
SA(Joe): 4(3.14)(.62)^2 = 4.83

V(Harry): 1/2 = 4/3(3.14)r^3, r = .49
SA (Harry): 4(3.14)(.49)^2 = 3.02

Relating time to surface area:
1/(40)4.83 = x/(40)3.02
x = 120.8/193.2 = .625 hours

converting to minutes
(60/1)(.625/x) = 37.5

```

```
Date: 09/11/2006 at 13:20:02
From: Doctor Peterson
Subject: Re: Physics speed comparison with varying surface areas and mass

Hi, Drew.

Your teacher has made an assumption that is not necessarily true; in
fact most of physics is based on making simplifying assumptions, and
it can be tricky to decide how far to go.  I'll explain what he
presumably did, then suggest reasons why it might be a wrong
assumption.

Peeling a potato is largely a matter of removing skin.  The amount of
skin on a potato is measured by the surface area.  Therefore, you can
guess that the time taken to peel will be proportional to the surface
area of the potato.

Each person peels about 40 potatoes.  The mass of one of Joe's is
twice that of Harry's, so (assuming that all potatoes have the same
density) the ratio of volumes is 2.  The volume of a solid is
proportional to the cube of the radius (or any linear measurement),
so if the potatoes are similar (the same shape but different sizes),
the ratio of the radii is the cube root of 2, 2^(1/3).  The surface
area is proportional to the square of the radius, so the ratio of
surface area is the square of the cube root of 2, 2^(2/3) = 1.587.
This is presumably also the ratio of the times to peel one potato,
and again, the ratio of the times to peel 40 potatoes.  So

Joe's time
------------ = 2^(2/3)
Harry's time

and
1 hour              60 min
Harry's time = ------- = 0.63 hr * ------ = 37.8
2^(2/3)              1 hr

If the "thoughts" are your answer, then you seem to be right, even
though I did what the teacher apparently meant to do.  Maybe he made
a mistake in the arithmetic?  I can't see a legitimate way to get 48
minutes for the answer.  It's possible he used a calculator to find 2^
(2/3) and entered it as 2^2/3, which would give 1.3333, making
Harry's time the reciprocal of that, or .75 hour = 45 minutes.  But
48 minutes would be 0.80 hour, and I don't see what error would give
that.

Your work is based on supposing the density to be 1, which is valid
because you can choose a unit in which it would be.  Everything is
still proportional.  The difference between your answer and mine is
probably from rounding.

Now, it's a good idea to think about what assumptions were made:

1. That peeling time is proportional to surface area.  This is
probably false, because a smaller potato might be harder to handle,
and more of the time will be spent picking up new ones (twice as
many times) rather than just peeling.

2. That the potatoes are similar, so that the proportions are valid.
This is possibly false: Large potatoes might be more elongated than
small ones, rather than being the same shape.

3. There are other assumptions, such as that potatoes have the same
density, and perhaps that their skin has the same thickness (or that
that does not affect the time for peeling).

Ignoring this difficulty, which in my mind makes the whole question
questionable, I think your answer is correct.  You should ask your
teacher for the details of his work, so we can verify that it is an
arithmetic error and not some fact we've missed.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Physics/Chemistry

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