Problem 106 - Chemistry Principles and Practice

Question

The U.S. debt in 2008 was \(\$ 9.2\) trillion. (a) Estimate the height, in kilometers, of a stack of 9.2 trillion \(\$ 1\) bills. Assume that a \(\$ 1\) bill has a thickness of \(0.166 \mathrm{~mm}\). (b) Estimate the mass of this stack if a \(\$ 1\) bill has a mass of \(1.01 \mathrm{~g}\)

Step-by-Step Solution

Answer

The stack is 1,527,200 km tall and weighs 9,292,000,000 kg.

Step 1: Convert Thickness to Kilometers

First, convert the thickness of a single \( \$1 \) bill from millimeters to kilometers. We are given that the thickness is \( 0.166 \) mm. Since there are 1,000,000 mm in a kilometer, we have:\[0.166 \text{ mm} = \frac{0.166}{1,000,000} \text{ km}.\]

Step 2: Calculate Total Height

Next, calculate the total height of 9.2 trillion \( \$1 \) bills stacked on top of one another. Multiply the number of bills, \( 9.2 \times 10^{12} \), by the thickness of each bill in kilometers:\[9.2 \times 10^{12} \times \frac{0.166}{1,000,000} \text{ km}.\]Substituting, we get:\[= 1,527,200 \text{ km}.\]

Step 3: Calculate Total Mass

Now, calculate the total mass of the stack of \( \$1 \) bills. Multiply the number of bills, \( 9.2 \times 10^{12} \), by the mass of each bill, \( 1.01 \text{ g} \):\[9.2 \times 10^{12} \times 1.01 \text{ g}.\]Convert this to kilograms, as there are 1000 grams in a kilogram:\[= \frac{9.292 \times 10^{12}}{1000} \text{ kg} = 9,292,000,000,000 \text{ kg}.\]

Key Concepts

Metric System

The Metric System is a universal system of measurement used worldwide. It is based on multiples of ten, making it easy to convert between units. For instance, in this exercise, we consider the thickness of a \\(1\\) bill given in millimeters (mm). To perform unit conversion in the metric system:

• Recognize that 1 kilometer (km) equals 1,000,000 millimeters.
• To convert millimeters to kilometers, divide by 1,000,000.

In the calculation:
A thickness of \0.166\ mm converts to kilometers using the formula:\[0.166 \, \text{mm} = \frac{0.166}{1,000,000} \, \text{km}\.\]This provides a seamless conversion, maintaining precision and clarity in distance measurements. Understanding these conversions is essential, not only in academics but also in practical daily life where metric measurements are prominent.

Scientific Notation

Scientific Notation is a method used to express very large or very small numbers. It simplifies calculations and reading by ensuring you only deal with manageable numbers. This concept is crucial in our exercise as we deal with trillions of bills. Here's why:

• Scientific notation helps express the U.S. debt of \(9.2\) trillion as \(9.2 \times 10^{12}\).
• It offers a clear and concise way to represent this enormous figure, which would be daunting if written out entirely.

In scientific notation, every number is written as the product of a number between 1 and 10, and a power of 10. For 9.2 trillion, we use\[9.2 \times 10^{12}\].This indicates that we've moved the decimal point 12 places to the right. It provides efficiency in calculations because when multiplying or dividing large numbers, you only need to adjust the exponents, reducing errors and complication.

Mathematical Estimation

Mathematical Estimation is a skill used to find approximate answers, making it valuable where exact answers are unnecessary or difficult to obtain. This primary skill helps simplify complex real-world problems like estimating the U.S. debt.In our exercise:

• We estimate the total height and mass of a stack of \9.2\ trillion \\(1\\) bills.
• The height is calculated by multiplying the number of bills by each bill's thickness (after conversion to kilometers).

This provides an estimate of the stack height at approximately 1,527,200 km. Likewise, estimating the mass involves multiplying the number of bills by the weight per bill and converting the total to kilograms.Estimation simplifies planning and decision-making by providing a ballpark figure when precise figures are not crucial. It is often used in budgeting, construction, and resource management, illustrating its importance beyond academic exercises.