How much tar paper?

My dad got a partially used role of tar paper and wanted to know how many square feet it has left.


The roll is 36.0 in. wide, with a 6.00 in. diameter.  The hollow core of the roll has a 2.88 in. diameter, and the tar paper is 0.02 in. thick.

Is there enough to cover 4 faces of an 8-ft. cube?

I like this problem because it can be solved in at least two ways.  What is the answer?

Tapering off my drug dependence

I had back surgery recently and have been taking the narcotic oxycodone, which has a half-life of 3.8 hours approximately.  I have been tapering my down my dose, taking

10mg / 4 hours

10 mg / 5 hours

10 mg / 6 hours

And now I want to drop to 5 mg, but wondered with what frequency I should take it.  I'll look at the levels remaining in my system after the doses given above, so I need to solve the half-life equation 0.5 = e^(3.8k), then evaluate

10e^(4k)

10e^(5k)

10e^(6k)

I get 4.83mg, 4.03mg, 3.36mg, respectively.  So when I drop down to the 5mg tablet, I'd like to take another when I have about 3.5mg left.  Let's see when that occurs:

3.5 = 5e^(kt)

Using k from above (have you found it yet?), I get t =~2 hours.  So I'll start with that period and increase from there.

Blank numbers

A few years ago I was teaching complex numbers in an algebra 2 class.  It took students awhile to grasp the idea of i.  I told them it was defined as the square root of negative one.

"But you can't square root a negative."

"That's why we have to create a new definition - we want to see what results from this definition," I said.

One student was particularly enamored by this idea, and I suggested he look at the other forbidden operation (besides square rooting a negative): dividing by zero.  He defined 1/0 as u, perhaps for "undefined," but called numbers involving u "blank numbers," since a calculator would return a blank screen after the error.

Over several days I prodded this student to create theorems about blank numbers, and to avoid contradictions by adding "special rules."  I have since lost his work, and don't remember the stumbling block that kept him from continuing.

Perhaps you can find it. Let 1/0 = u.

Thm. 1: u² = u

Where does it go from here?

When does order matter?

Maybe Tammy was thinking about matrices?

Sketchy recursion in Inception

In the movie Inception, most of which takes place inside people's dreaming minds, we are told that 5 minutes of dreaming equates to 1 hour inside that dream (a 1:12 ratio) because "your mind functions more quickly."  If, once you are dreaming, you go to sleep and have a dream (within a dream), the same ratio holds, so that 5 minutes in the "real world" equates to 12 hours in the 2nd-layer dream.

We are told that the more layers of dreams-within-dreams you have, the more unstable your situation becomes, so the pharmacist administers some powerful sedatives to help.  But this changes the ratio from 1:12 to 1:20 somehow, and now 5 real minutes should give you 100 minutes in the first level of dreaming, 2000 minutes (33) hours in the 2nd level, and so on.

The numbers given in the movie were a bit sketchy, though - instead of "100 minutes, 33 hours, 28 days," they said something like "2 hours, 1 week, 6 months," or something like that.

The other fishy math was the amount of dreaming that got done during the time the van was falling from the bridge to the water.  If we're generous and say the bridge was 100 ft above the water, it should have been about a 4.5-second fall.  The should have given them 90 seconds in the hotel and 30 minutes in the snowy wonderland.  Max.

Do I have this right?