Approximating Vast Numbers: a Trillion Dollars

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.

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
   $$\begin{array}{l}\text{volume of stack}=\text{length}\times \text{width}\times \text{height,}\\ \text{volume of stack}=\text{6 in}\text{.}\times \text{3 in}\text{.}\times \text{0}\text{.5 in}\text{.,}\\ \text{volume of stack}=\text{9 in}{\text{.}}^{\text{3}}.\end{array}$$

2. Calculate the number of stacks. Note that a trillion dollars is equal to $\text{\$}1\times {10}^{12}$, and a stack of one-hundred $\text{\$}100$ bills is equal to $\text{\$}10,000,$ or $\text{\$}1\times {10}^4$ . The number of stacks you will have is

   $${\text{\$1 × 10}}^{\text{12}}{\text{ (a trillion dollars) / \$1 × 10}}^{\text{4}}{\text{ per stack = 1 × 10}}^{\text{8}}\text{ stacks}\text{.}$$

   1.3

3. Calculate the area of a football field in square inches. The area of a football field is $100\text{ yd}\times \text{ 50}\text{yd}$, which gives $5,000{\text{ yd}}^{\text{2}}$ . Because we are working in inches, we need to convert square yards to square inches
   
   

   $$\begin{array}{l}{\text{Area = 5,000 yd}}^{\text{2}}\text{ × }\frac{\text{3 ft}}{\text{1yd}}\text{ × }\frac{\text{3 ft}}{\text{1yd}}\text{ × }\frac{\text{12 in}\text{.}}{\text{1 foot}}\text{ × }\frac{\text{12 in}\text{.}}{\text{1 foot}}\text{ = 6,480,000 in}{\text{.}}^{\text{2}}\text{,}\hfill \\ \text{Area}\approx {\text{6 × 10}}^{\text{6}}\text{ in}{\text{.}}^{\text{2}}\text{.}\hfill \end{array}$$

   This conversion gives us ${\text{6 × 10}}^{\text{6}}\text{ in}{\text{.}}^{\text{2}}$ 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 $$\text{9 in}{\text{.}}^{\text{3}}{\text{ / stack × 10}}^{\text{8}}{\text{ stacks = 9 × 10}}^{\text{8}}\text{ in}{\text{.}}^{\text{3}}$$
5. Calculate the height. To determine the height of the bills, use the following equation
   $$\begin{array}{ccc}\text{volume of bills}& \text{=}& \text{area of field × height of money}\\ \text{Height of money}& \text{=}& \frac{\text{volume of bills}}{\text{area of field}}\\ \text{Height of money}& \text{=}& \frac{{\text{9 × 10}}^{\text{8}}\text{ in}{\text{.}}^{\text{3}}}{{\text{6 × 10}}^{\text{6}}\text{ in}{\text{.}}^{\text{2}}}\text{ = 1.5}{\text{ × 10}}^{\text{2}}\text{ in}\text{.}\end{array}$$

   Since the area is actually a little larger than 6 × 10², the height is a little less than 150, so round down to 100 inches rather than up to 200 inches.

   $$\text{Height of money}={\text{1×10}}^{\text{2}}\text{ in.}=\text{100 in}\text{.}$$

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

   $$100\text{ in}\text{.}\times \frac{\text{1ft}}{\text{12 in}\text{.}}=8.33\text{ ft}\approx \text{8 ft}\text{.}$$