Three tricky problems

[sallycats]

My teacher says that these tricky problems are "counterintuitive". I'm having a very difficult time getting started. If anyone has any hints or clues as to where I should start on these, please let me know.

1. The big gear has 24 teeth and cannot turn. The little gear with 8 teeth is free to
roll around it. When the little gear makes one full trip around the big gear, how many somersaults will
the letter A make?

2. Imagine a metal band wrapped snugly around the equator of a perfectly smooth planet the
size of Earth. If the band were lengthened by 12 inches, it wouldn’t be as snug. If the earth and the
lengthened band are concentric, what is the distance between the band and the planet?

3. What is the probability that there are two people in a city with population 3,000,000 who have exactly the
same number of hairs on their heads? (Exclude totally bald people from consideration!)

(apologies if I'm posting this in the wrong forum section).

 

[mmm4444bot]

Hi sallycats:

In the first exercise, there is not a one-to-one correspondence between a single revolution of the small gear and a single somersault of the letter A. In other words, the letter A will complete its first somersault before the small gear makes one revolution.

If you have trouble visualizing this, print out two enlarged copies of the diagram, and cut out the small gear from one sheet. Number the cogs on this cut-out from 1 through 8. Number the cogs on the big gear from 1 through 24.

Now, rotate the small gear about the big gear, cog by cog, to get a sense of what's happening.

In the second exercise, use the formula for the circumference of a circle to calculate the circumference of the circle at the equator. If you want to work with actual numbers, a commonly-used mean radius for planet Earth is 3959 miles. This will give the circumference of a "smaller" circle (in miles).

Next, create a "larger" circle, by adding 12 inches to the circumference of the "smaller" circle. You'll first need to convert 12 inches into miles because we can only add miles to miles.

Go back to the circumference formula, and calculate the radius of this "larger" circle.

The exercise asks for the difference between the two radii. You can then convert back to inches.

Alternatively, you can go through the same steps using symbolic radii (like r for the small circle and R for the big circle). Then all of the units will be inches, and you won't need to do any conversions.

In the third exercise, I'm thinking that we would need to begin by stating some assumptions about the mean and standard deviation of hair numbers, in that city of three million people, but (off the top of my head, heh, heh) I'm not sure about that.

Cheers ~ Mark

 

[soroban]

Hello, sallycats!

2. Imagine a metal band wrapped snugly around the equator of a perfectly smooth planet the size of Earth.
If the band were lengthened by 12 inches, it wouldn’t be as snug.
If the earth and the lengthened band are concentric, what is the distance between the band and the planet?

Jeez -- don't mess around with miles, feet and inches!

The radius of the earth is: \(\displaystyle R\).
The circumference of the earth is: \(\displaystyle C \:=\:2\pi R\)

If the radius is increased by some amount, \(\displaystyle d\)
. . the new radius would be: .\(\displaystyle R + d\)
. . and the new circumference would be: .\(\displaystyle C' \:=\:2\pi(R + d)\)

We're told that the difference is 12 inches.

. . \(\displaystyle 2\pi(R+ d) - 2\piR \:=\:12 \quad\Rightarrow\quad 2\pi R + 2\pi d - 2\pi R \:=\:12\)

\(\displaystyle \text{Hence: }\;2\pi d \:=\:12} \quad\Rightarrow\quad d \:=\:\frac{12}{2\pi} \:=\:\frac{6}{\pi} \:=\:0.954929659\text{ inches.}\)

The band is nearly an inch above the ground.

 

[mmm4444bot]

Jeez -- don't mess around with [numbers and units]

Jeez -- with "hints or clues as to where [to] start" like soroban's, there really is no need to "mess around" at all. The rest of us can go watch Hulu, instead.

 

[Mrspi]

Re:

Jeez -- don't mess around with [numbers and units]

Jeez -- with "hints or clues as to where [to] start" like soroban's, there really is no need to "mess around" at all. The rest of us can go watch Hulu, instead.

That kind of "help" takes the challenge out for the student...and the LEARNING, too. ARGH!!!!

 

[sallycats]

Thanks everyone for your help!
I went through the problem using Mark's advice first, and then reviewed Soroban's post to check my work. No Hulu for me :wink:
However, I think 6/? = 1.909859317, not 0.9549..