Example 2: Approximating Vast Numbers: a Trillion Dollars

The U.S. federal deficit in the 2008 fiscal year was a little greater 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?

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:

[latex]\begin{align*} \text{Volume of stack} &= \text{length} \times \text{width} \times \text{height} \\ &= 6~\text{in} \times 3~\text{in} \times 0.5~\text{in} \\ &= 9~{\text{in}}^3 \end{align*}[/latex]

(2) Calculate the number of stacks. Note that a trillion dollars is equal to $1 × 10¹², and a stack of one-hundred $100 bills is equal to $10,000, or $1 × 10⁴. The number of stacks you will have is:

[latex]\dfrac{$1 \cdot 10^{12}~\text{(a trillion dollars)}}{1} \cdot \dfrac{1~\text{stack}}{$1 \cdot 10^4}=1 \cdot 10^8~\text{stacks}[/latex].

(3) Calculate the area of a football field in square inches. The area of a football field is 100 yd × 50 yd, which gives 5000 yd² Because we are working in inches, we need to convert square yards to square inches:

[latex]\begin{align*} \text{Area}&=5000~{\text{yd}}^2 \times \dfrac{3~\text{ft}}{1~\text{yd}} \times \dfrac{3~\text{ft}}{1~\text{yd}} \times \dfrac{12~\text{in}}{1~\text{ft}} \times \dfrac{12~\text{in}}{1~\text{ft}}\\[2mm] &=6,480,000~{\text{in}}^2 \end{align*}[/latex]

This conversion gives us 6 × 10⁶ 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

[latex]\dfrac{9~{\text{in}^3}}{1~\text{stack}} \times 10^8 stacks=9 \cdot 10^8~{\text{in}}^3[/latex].

(5) Calculate the height. To determine the height of the bills, use the equation:

[latex]\begin{align*} \text{Volume of bills} &= \text{Area of field} \times \text{Height of bills} \\[2mm] \text{Height of bills} &= \dfrac{\text{Volume of bills}}{\text{Area of field}} \\[2mm] &= \dfrac{9 \cdot 10^8~{\text{in}}^3}{6 \cdot 10^6~{\text{in}}^2} = 1.33 \cdot 10^2~\text{in} \\[2mm] \text{Height of bills} &\approx 1 \cdot 10^2~\text{in} = 100~\text{in} \end{align*}[/latex]

The height of the money will be about 100 inches high. Converting this value to feet gives

[latex]100~\text{in} \times \dfrac{1~\text{ft}}{12~\text{in}} = 8.33~\text{ft} \approx 8~\text{ft}[/latex]

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?