At the time of this book's printing, the U.S. national debt...

At the time of this book's printing, the U.S. national debt is about $\$ 6$ trillion. (a) If payments were made at the rate of $\$ 1000$ per second, how many years would it take to pay off the debt, assuming no interest were charged? (b) A dollar bill is about $15.5 \mathrm{cm}$ long. If six trillion dollar bills were laid end to end around the Earth's equator, how many times would they encircle the planet? Take the radius of the Earth at the equator to be $6378 \mathrm{km} .$ (Nole: Before doing any of these calculations, try to guess at the answers. You may be very surprised.)

Key Concepts

Estimation and Approximation

Estimation involves making an educated guess about the value of a quantity to simplify calculations and verify results. This technique is useful for obtaining a quick sense of magnitude and ensuring that detailed calculations are on the right track.

Rate Problems

Rate problems involve calculations where a quantity accumulates or diminishes over time at a constant rate. Understanding the relationship between rate, time, and quantity enables one to determine, for example, how long it will take to complete a task given a fixed rate of progress.

Geometry of Circles (Circumference Calculation)

The concept of the circumference of a circle, calculated using the formula 2? times the radius, is fundamental in determining distances around circular shapes. This is particularly useful when comparing linear measures, such as a string of objects, to the roundness of the Earth.

Unit Conversion

Converting units is essential when working with different measurements in a problem. It involves changing quantities from one unit to another, such as seconds to years or centimeters to kilometers, ensuring that all parts of a calculation are expressed in compatible units.

Scientific Notation and Large Numbers

Scientific notation is a method of expressing very large or very small numbers in a concise form by using powers of ten. This allows for easier manipulation, comparison, and estimation of values that would otherwise be cumbersome to handle in everyday arithmetic.

Transcript

00:01 Okay, we are $6 trillion in debt.

00:08 Six trillion.

00:09 That's actually six times 10 to $12 in depth.

00:15 And our rate is $1 ,000 per second.

00:23 We need to figure out how many years it takes to meet this debt.

00:30 So we'll just do our usual conversions.

00:34 So we do $1 ,000 per second times 60 seconds in one minute.

00:43 The seconds cancel out.

00:47 There's 60 minutes in one hour.

00:52 The minutes cancel out.

00:55 There's 24 hours in one day.

01:00 The hours cancel out.

01:01 And finally, there's 365 days in one year.

01:08 And the days cancel out.

01:14 And we multiply all of these together, and we would get 3 .15 times 10 to the $10.

01:25 Per one year okay so let's see how many years it would take all you have to do is just take the debt so six times 10 to 12 the six trillion dollars and you divide it by our rate so 3 .1 divided by 3 .15 times 10 to the 10 and finally we would have our years so it would take 190 years to this debt.

02:12 Okay, now on to the next part.

02:15 Let's figure out how many times six trillion dollars bills laid end to end go around the equator of the earth.

02:25 A pretty peculiar question, but let's let's do it.

02:30 So first things first, let's figure out the circumference of the earth, which is 2 pi, c .e.

02:39 Equals 2 pi times the radius, which is right here, times 6 ,3007 .7 .5 .5 .5 .5 .5.

02:47 28 kilometers and this equals to 4 ,000, 474 kilometers...

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