Question

Indicate the number of significant figures in each of the following: a. This book contains over 500 pages. b. A mile is just over \(5000 \mathrm{ft}\). c. A liter is equivalent to 1.059 qt. d. The population of the United States is approaching 250 million. e. A kilogram is \(1000 \mathrm{~g}\). f. The Boeing 747 cruises at around \(600 \mathrm{mph}\).

Short answer

a. 1 significant figure b. 1 significant figure c. 4 significant figures d. 2 significant figures e. 1 significant figure f. 1 significant figure

Step-by-step solution

a. 500 pages

The given number "500" has one significant figure as the two trailing zeros do not contain a decimal point.

b. 5000 ft

The given number "5000" has one significant figure as the three trailing zeros do not contain a decimal point.

c. 1.059 qt

The given number "1.059" has four significant figures as it contains no leading or trailing zeros, and all digits are non-zero.

d. 250 million

The given number "250" has two significant figures as the trailing zero does not contain a decimal point. Note that the word "million" does not affect the number of significant figures.

e. 1000 g

The given number "1000" has one significant figure as the three trailing zeros do not contain a decimal point.

f. 600 mph

The given number "600" has one significant figure as the two trailing zeros do not contain a decimal point.

Key concepts

Scientific Notation

Scientific notation is a helpful way to express very large or very small numbers. It is particularly useful in fields such as physics and chemistry where these kinds of numbers are common.
In scientific notation, a number is written as a product of a number, called the coefficient, and a power of ten. For example, the number 5000 can be expressed as \(5 \times 10^3\). This makes it much easier to work with during calculations.
- **One main benefit** is that it automatically indicates the number of significant figures. - **In scientific studies**, it allows scientists to easily manipulate and compare these values without losing track of precision.
When writing in scientific notation:

• Only include significant figures in the coefficient.
• The number of places the decimal point has moved provides the exponent of 10.

Measurement Precision

Measurement precision refers to how repeatable a measurement is. In simple terms, it addresses the question: "If I measure something multiple times, do I get the same result each time?"
- **Precision is essential**, especially in scientific experiments. - If a measurement is precise, it has little variation between repeated measurements.
Measurement precision is deeply linked to significant figures. The more significant figures, the more precise the measurement is assumed to be. Having precise measurements can elevate the reliability of results and conclusions drawn from them.
Here are a couple of reasons precision is important:

• Exact results help in identifying trends and patterns.
• It ensures that scientific research is credible and can be reproduced.
• High precision minimizes potential errors in calculations and interpretations.

Numerical Accuracy

Numerical accuracy differs slightly from precision. Accuracy is about how close a measurement is to the true value or accepted standard. It's possible to be accurate without being precise, or precise without being accurate.
For instance, if you weigh a 1kg object, a result of 1.05kg might be accurate if that's correct within the device's range. Still, the measurements need to be both accurate and precise for the best results in scientific studies.
When discussing accuracy and significant figures:

• Greater significant figures often contribute to increased accuracy.
• Ensures consistency in data representation across different studies and disciplines.

Numerical accuracy is crucial as it offers:

• A solid basis for data comparison and analysis.
• Ensures that findings reflect true measurements and conditions.