Question

Indicate the number of significant figures in each of the following measured quantities: (a) \(62.65 \mathrm{~km} / \mathrm{hr}\), (b) \(78.00 \mathrm{~K}\), (c) \(36.9 \mathrm{~mL}\) (d) \(250 \mathrm{~mm}\), (e) 89.2 metric tons, (f) \(6.4224 \times 10^{2} \mathrm{~m}^{3}\)

Short answer

(a) 4 significant figures, (b) 4 significant figures, (c) 3 significant figures, (d) 2 significant figures, (e) 3 significant figures, (f) 5 significant figures.

Step-by-step solution

:For the given quantity, all the digits (6, 2, 6, and 5) are non-zero and there are no zeros between non-zero digits. Hence, all the digits are considered significant. Therefore, there are 4 significant figures in \(62.65 \mathrm{~km}/\mathrm{hr}\). ##Step 2: Determine the significant figures in (b) \(78.00 \mathrm{~K}\).##

:In this case, we have two non-zero digits (7 and 8) and two trailing zeros after the decimal point (0 and 0). Since the number has a decimal point, the trailing zeros are significant. Thus, there are 4 significant figures in \(78.00 \mathrm{~K}\). ##Step 3: Determine the significant figures in (c) \(36.9 \mathrm{~mL}\).##

:Here, all the digits (3, 6, and 9) are non-zero, and there are no zeros between non-zero digits. Therefore, there are 3 significant figures in \(36.9 \mathrm{~mL}\). ##Step 4: Determine the significant figures in (d) \(250 \mathrm{~mm}\).##

:For this quantity, we have two non-zero digits (2 and 5) and one trailing zero without a decimal point (0). The trailing zero is not considered significant since there is no decimal point. Thus, there are 2 significant figures in \(250 \mathrm{~mm}\). ##Step 5: Determine the significant figures in (e) 89.2 metric tons.##

:In this case, all the digits (8, 9, and 2) are non-zero. There are no zeros between non-zero digits. Therefore, there are 3 significant figures in 89.2 metric tons. ##Step 6: Determine the significant figures in (f) \(6.4224 \times 10^{2} \mathrm{~m}^{3}\).##

:The number is in scientific notation, which means that only the significant digits are written (6, 4, 2, 2, and 4). There are no zeros between them, so there are 5 significant figures in \(6.4224 \times 10^{2} \mathrm{~m}^{3}\). To summarize the number of significant figures in each measured quantity: (a) 4 significant figures, (b) 4 significant figures, (c) 3 significant figures, (d) 2 significant figures, (e) 3 significant figures, (f) 5 significant figures.

Key concepts

Measured Quantities

Measured quantities are values that we obtain through the process of measurement. These numbers are not exact because they involve using measuring tools, which can have varying degrees of accuracy. Understanding measured quantities helps us engage with real-world data in a meaningful way. Measurement results are expressed with significant figures to indicate their precision.
It is important to be aware that each digit in a measured quantity carries importance in expressing the data's precision:

• Each non-zero digit is significant, helping us understand the accuracy of the measurement.
• Zeros may or may not be significant, depending on their position in the number.

Recognizing the significant figures in measured quantities is a core skill in scientific and mathematical work as it reflects the precision of our tools and the limitations of our data.

Scientific Notation

Scientific notation is a compact way of writing very large or very small numbers. This notation is especially common in scientific and technical fields where clarity with large calculations is essential. An example in the original exercise is the quantity given as \(6.4224 \times 10^{2} \text{ m}^3\).
The format for scientific notation is:

• A decimal part, which includes the significant figures of the number, such as 6.4224.
• A base of ten raised to an exponent, indicating how many places the decimal point has been moved.

Using scientific notation allows numbers to be more easily read and clearly presented, avoiding oversights in mathematical expressions.
This method is also highly beneficial when indicating significant figures, as only the significant digits are included in the decimal part, streamlining the expression of precision.

Significant Digits

Understanding significant digits is crucial when working with measured quantities. They indicate the reliability of a measurement by showing which digits are meaningful in conveying detail about the measure. Let's break down the rules for identifying them:

• All non-zero numbers are always significant because they indicate actual measured values.
• Any zeros between non-zero digits are considered significant. For instance, in the number 107.02, all digits are significant.
• Leading zeros before a number are not significant; they merely indicate the position of the decimal point.
• Trailing zeros are significant only if a decimal point is present. For example, in the number 78.00 K, the zeros are significant because of the decimal point.

These rules help you identify the number of significant figures in any given measurement, allowing you to accurately report and interpret data.