Example 1.4 from College Physics, 1.4 Approximation

Approximating Vast Numbers: a Trillion Dollars

A bank stack containing one hundred hundred-dollar bills.

Figure 1.27 A bank stack contains one-hundred $100 bills, and is worth $10,000. How many bank stacks make up a trillion dollars? (credit: Andrew Magill)

The U.S. federal debt in the 2008 fiscal year was a little less than $10 trillion. Most of us do not have any concept of how much even one trillion actually is. Suppose that you were given a trillion dollars in $100 bills. If you made 100-bill stacks and used them to evenly cover a football field (between the end zones), make an approximation of how high the money pile would become. (We will use feet/inches rather than meters here because football fields are measured in yards.) One of your friends says 3 in., while another says 10 ft. What do you think?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
When you imagine the situation, you probably envision thousands of small stacks of 100 wrapped $100 bills, such as you might see in movies or at a bank. Since this is an easy-to-approximate quantity, let us start there. We can find the volume of a stack of 100 bills, find out how many stacks make up one trillion dollars, and then set this volume equal to the area of the football field multiplied by the unknown height.
Solution
(1) Calculate the volume of a stack of 100 bills. The dimensions of a single bill are approximately 3 in. by 6 in. A stack of 100 of these is about 0.5 in. thick. So the total volume of a stack of 100 bills is:

$\begin{array}{l} volume of stack = length \times width \times height, \\ volume of stack = 6 in . \times 3 in . \times 0 . 5 in ., \\ volume of stack = 9 in .^{3} . \end{array}$
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(2) Calculate the number of stacks. Note that a trillion dollars is equal to $$ 1 \times 10^{12},$ and a stack of one-hundred $$ 100$ bills is equal to $$ 10, 000,$ or $$ 1 \times 10^{4}$ . The number of stacks you will have is:
$$ 1 \times 10^{12} (a trillion dollars) / $1 \times 10^{4} per stack = 1 \times 10^{8} stacks.$
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(3) Calculate the area of a football field in square inches. The area of a football field is $100 yd \times 50 yd,$ which gives $5, {000 yd}^{2} .$ Because we are working in inches, we need to convert square yards to square inches:
$\begin{matrix} Area = {5,000 yd}^{2} \times \frac{3 ft}{1 yd} \times \frac{3 ft}{1 yd} \times \frac{12 in .}{1 ft} \times \frac{12 in .}{1 ft} = 6, 480, 000 in .^{2}, \\ Area \approx 6 \times 10^{6} in .^{2} . \end{matrix}$
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This conversion gives us $6 \times 10^{6} in .^{2}$ for the area of the field. (Note that we are using only one significant figure in these calculations.)
(4) Calculate the total volume of the bills. The volume of all the $$ 100$ -bill stacks is $9 in .^{3} / stack \times 10^{8} stacks = 9 \times 10^{8} in .^{3} .$
(5) Calculate the height. To determine the height of the bills, use the equation:

$\begin{array}{l} volume of bills & = & area of field \times height of money: \\ Height of money & = & \frac{volume of bills}{area of field}, \\ Height of money & = & \frac{9 \times 10^{8} in .^{3}}{6 \times 10^{6} {in.}^{2}} = 1.33 \times 10^{2} in., \\ Height of money & \approx & 1 \times 10^{2} in. = 100 in. \end{array}$
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The height of the money will be about 100 in. high. Converting this value to feet gives

$100 in . \times \frac{1 ft}{12 in .} = 8 . 33 ft \approx 8 ft.$
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Discussion
The final approximate value is much higher than the early estimate of 3 in., but the other early estimate of 10 ft (120 in.) was roughly correct. How did the approximation measure up to your first guess? What can this exercise tell you in terms of rough “guesstimates” versus carefully calculated approximations?

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