1.4 Approximation  (Page 2/3)

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:

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

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

This conversion gives us $6\times {\text{10}}^6\text{in}{\text{.}}^2$ 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 $\$\text{100}$ -bill stacks is $9\text{in}{\text{.}}^3/\text{stack}\times {\text{10}}^8\text{ stacks}=9\times {\text{10}}^8\text{in}{\text{.}}^3.$

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

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

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?

Summary

Scientists often approximate the values of quantities to perform calculations and analyze systems.

Problems&Exercises