Simple conversions and calculations

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Marooned and far from home

In Cast Away, Tom Hanks was marooned on an uninhabited Pacific island. In the scene we have viewed he computes the search area that rescuers must search. So, we have to determine the search region and its area. Tom assumes the search area is a circle centered at the point of last radio contact. He assumes the cargo jet was traveling at 400 mph.

We see Tom make the calculation of Area = $\pi \times ~$(radius)$^{2}$:

$160000$ $\unit{mi}$ This is MATH
$\times \ \ 3.14$ Is his calculation correct?
$502,400$ $\unit{mi}$ ????

Using the famous formula for the area, of a circle, $A=\pi r^{2}, $ we have the exact calculation

Tom is really very close; Tom knows his math. Notice how he convinces all of us how vast the area is by saying its "twice the size of Texas."

Approximations to $\pi $ have been studied for thousands of years. Archimedes gave us

from which we obtain MATH our decimal approximation $3.14.$ Ludolph van Ceulen (1540-1610) computed $\pi $ accurate to 31 places. So proud was he of this accomplishment that he had the digits engraved on his tombstone.

What is known is that $\pi $ is a transcendental number and, of course, doesn't have a repeating decimal expansion. So, just how many digits of $\pi $ are known? Answer: 6,000,000,000 (6 billion) digits? You can download them all.

Other important area formulas are:




How far can you run in 90 minutes?

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In the movie, The Fugitive, our hero Dr. Richard Kimball (Harrison Ford) is trying to save his reputation and catch the murderer. Before that happens, he must escape from prison. The Deputy US Marshall Samuel Gerard,played by Tommy Lee Jones, is bent on putting Kimball back in jail. First he must find him. When he arrives at the sceen of the wreck,

he says, "Our fugitive has now been on the run for 90 minutes. Foot speed over uneven ground is four miles per hour. That gives us a radius of six miles." Is this correct?

How did he make the computation? It is a units conversion problem and a rate equation.

To use the rates equation we must have common units. Our data:

Substituting gives 4 miles/hour $\times $ 90 minutes = ??? First convert minutes to hours:
Now substitute to get:

Deputy Gerard knows his math.


How much gold can is worth the effort?

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In Kelly's Heroes, Kelly (played by Clint Eastwood) wants to raid a German bank behind enemy lines. To get supplies he tempts a couple of sargents with bars of gold. Here we see Crapgame with just one bar of gold. From just this and a quotation of the price of gold on the Paris commondity exchange, and the fact that there are 14,000 bars of gold, he computes the value to be $16,000,000. Is he right?

Compiling our data. Let us suppose a bar of gold is 30cm $\times $ 5cm $\times $ 2cm. Then

To get the value of one bar of gold we multiply:

But there are other conversions:MATH and MATH. To get the value of all the gold we multiply by the number of bars:


The estimate by Crapgame was way, way off. More data:

Volume: MATH

$\allowbreak $Weight/bar:MATH

Total Weight: MATH

In today's dollars, how much is the gold worth?


The case of the subway train

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In 12 Angry Men a young man is accused of murdering his father. All jurors but one want to convict him immediately. One juror stands up for the young man. During the course of the deliberations, the jurors discuss how long a six car elevated subway would take to pass the window of the victim. "It takes about 10 seconds," he said. Other jurors agreed. Does this value hold up?


Let us suppose that

The subway is MATH We also have
Thus the subway travels its length of MATH feet in
This is much, much less than the "guessed" 10 seconds. If the elevated train was moving at 45 miles/hr, the time would be less than 3 seconds. So, the jury is way off the mark. How many people have been convicted or released because of incorrect computations?????

Note: Modern subway cars are about 75 feet long. However, this movie was first released in 1955 and the version above in 1997, but with essentially the same script.


Exercises. Now its your turn.

  1. A snail moves about 3 cm/minute. Find the snail's speed in miles per hour. Find the snail's speed in furlongs per fortnight. (There are 8 furlongs to a mile and a fortnight is about two weeks.)

  2. A football player runs at 4 miles/minute in the dining car in the opposite direction of a train moving 30 miles/hours. How fast is the football player running relative to person standing on the road next to the railway tracks.

  3. Gold sells for $12/gram. Mary converts her weekly allowance of 10 pounds of gold to silver. How much is her allowance in dollars, and how much does the silver weigh? [The density of silver is MATH]

  4. In today's dollars, how much is Kelly's gold worth? Remember, the is the factor of inflation (a dollar was worth far more then) and the current price of gold is more than 10 times what it was in 1943, the time this movie takes place.

  5. What is the radius of the smallest circle that contains all of Texas? [Hint. Find the diameter of Texas. Is this the answer?]

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