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# Section 1.3 Large Variations are Made of Tiny Variations

### Light travels large distances in small bits

The speed of light is $2.998 \cdot 10^8$ meters/sec. This means that it travels $2.998 \cdot 10^8$ m in one second (one light-second).

However, light travels $2.998 \cdot 10^8$ m by traveling one meter at a time. So, it takes $\dfrac{1}{2.998} \cdot 10^{-8} \mathrm{sec}$, which is $3.336 \cdot 10^{-9}$(3.3 billionths) sec to travel one meter.

Light also travels one light-second by traveling one millimeter at a time. Light therefore would travel one millimeter, which is $\frac{1}{1000}$ meters ($1 \cdot 10^{-3}$ m), in $\frac{1}{1000}$ the time that light takes to travel one meter.

In the same vein, light would travel one micrometer ($1 \cdot 10^{-6}$ m) in $\frac{1}{1000}$ the time it takes to travel one millimeter. It would therefore take approximately $3.336 \cdot 10^{-15}$ seconds for light to travel one micrometer. (Check this for yourself.)

But light nevertheless travels one light-second in one second, or one light-year in one year, or a billion light-years in a billion yearsâ€”all the time traveling one micrometer at a time! It just travels a very large number of micrometers in a very large number of really small bits of time.

## Exercise Set 1.3

Use GC for your computations. Enter "*" for multiplication, "/" for division, shift-6 ("^") for an exponent, "$\downarrow$" to exit an exponent, "$\rightarrow$" to move to the next term.

You might be tempted to use your personal calculator for these exercises. PLEASE DON'T DO THIS. Use GC. It is by using GC that you will become comfortable using it. Being comfortable with GC will be very important for later chapters.

1. The Grand Canyon is enormous. It is 433 km long and has an average depth of 1.6 km. The US Forest Service estimates its volume at 4.17 trillion $(4.17 \cdot 10^{12}) \mathrm{m^3}$. Suppose the dump trucks that re-created the Sahara Desert (Exercise 1.1.5) are re-purposed, lining up to fill the Grand Canyon. Suppose also that they can dump one load every 30 seconds. How many years would it take to fill the Grand Canyon.
2. The dimensions of a dollar bill are 15.5956 cm by 6.6294 cm by 0.010922 cm. The U.S. federal government spent $\$3.8$trillion in fiscal year 2011. 1. How high, in km, would be a stack of dollar bills that contains the same amount of money as spent by the U.S. federal government in 2011? 2. An Olympic swimming pool is 50 m long, 25 m wide, and 1.5 m deep. How many Olympic pools would be filled by the dollar bills spent by the U.S. federal government in 2011? 3. A$\$100$ bill has the same dimensions as a $\$1$bill. An episode of the television program Breaking Bad shows three wooden platforms, each holding a stack of$\$100$ bills measuring approximately $1\, \mathrm{m^3}$. What, approximately, would be the value of those three stacks if they contained real $\$100\$ bills?
3. A sheet of A4 paper is 21 cm by 29.7 cm and is 0.1 cm thick. Suppose you can slice the sheet without losing any paper.

Slice it in half lengthwise and stack the slices. You will have a stack that is 0.2 cm high. The slices will be 14.85 cm by 21 cm.

Slice each of these pieces in half lenghwise and stack the pieces. You will have a stack 0.4 cm high. The slices will be 14.85 cm by 10.5 cm.

Continue this process a total of 50 times. How high will the stack be in km? In trips to the sun? What will be the dimensions of the slices?

4. Use these facts to determine how many miles the earth travels around the sun in the blink of an eye.
• The blink of an eye lasts approximately 1/3 second.
• The earth's orbit around the sun is approximately circular (it is slightly elliptical) with an average radius of 93 million miles.
• The earth completes one orbit in approximately 365.25 days.

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