Question

Indicate the number of significant figures in each of the following measured quantities: (a) \(3.774 \mathrm{~km}\), (b) \(205 \mathrm{~m}^{2}\), (c) \(1.700 \mathrm{~cm}\), (d) \(350.00 \mathrm{~K}\), (e) \(307.080 \mathrm{~g}\).

Short answer

The number of significant figures in the given measured quantities are: (a) \(3.774 \mathrm{~km}\) has \(4\) significant figures, (b) \(205 \mathrm{~m}^{2}\) has \(3\) significant figures, (c) \(1.700 \mathrm{~cm}\) has \(4\) significant figures, (d) \(350.00 \mathrm{~K}\) has \(5\) significant figures, and (e) \(307.080 \mathrm{~g}\) has \(6\) significant figures.

Step-by-step solution

(a) Significant Figures for 3.774 km

All non-zero digits are significant, and any zeros that are between non-zero digits are significant as well. Here, there are three non-zero digits (3, 7, 7) and one zero, but it is between two non-zero digits. Therefore, there are \(4\) significant figures in \(3.774 \mathrm{~km}\).

(b) Significant Figures for 205 m²

In this case, there are two non-zero digits (2 and 5) and one zero at the end. The trailing zero in this case is also considered significant because it indicates the precision of the measurement. Hence, there are \(3\) significant figures in \(205 \mathrm{~m}^{2}\).

(c) Significant Figures for 1.700 cm

Here, there are two non-zero digits (1 and 7) and two trailing zeros after the decimal point. The decimal point signifies that the zeros are significant, indicating the precision of the measurement. Therefore, there are \(4\) significant figures in \(1.700 \mathrm{~cm}\).

(d) Significant Figures for 350.00 K

In this case, there are two non-zero digits (3 and 5) and two zeros after the decimal point. Since the measurement has a decimal point, the trailing zeros are considered significant, indicating the precision of the measurement. Hence, there are \(5\) significant figures in \(350.00 \mathrm{~K}\).

(e) Significant Figures for 307.080 g

Here, there are four non-zero digits (3, 0, 7, and 8) and one zero between non-zero digits. Additionally, there is one trailing zero after the decimal point. As previously stated, the decimal point indicates that the trailing zero is significant. Therefore, there are \(6\) significant figures in \(307.080 \mathrm{~g}\).

Key concepts

Measurement Precision

Measurement precision refers to the closeness of repeated measurements of the same quantity to each other. Think of it as a way of expressing how finely tuned a measurement is. For instance, if you were to use a ruler that has millimeter markings instead of centimeter markings, your measurement would be more precise simply because you have a finer scale to work with.

To convey precision in a numeric form, scientists and mathematicians use significant figures. This system allows anyone reading a measurement to understand just how exact the figures are meant to be. For example, when we say that a room is 5.00 meters wide, the two zeros after the decimal point tell us that the measurement is precise to within one hundredth of a meter. Thus, in scientific and engineering calculations, tracking the precision of measurements is crucial to ensure that final results are reported with the appropriate level of certainty.

Scientific Notation

Scientific notation is a convenient way to express very large or very small numbers. It consists of two parts: a coefficient that is a number greater than or equal to 1 but less than 10, and an exponent that is a power of 10. This method not only makes it easier to read and write such numbers but also simplifies calculations and clearly indicates the number of significant figures.

For example, the number 0.0003056 can be written as \(3.056 \times 10^{-4}\) in scientific notation. This helps to immediately identify the number of significant figures, which in this case would be four (3, 0, 5, and 6). It's a powerful tool that, when combined with an understanding of significant figures, allows for precision and accuracy to be maintained throughout complex calculations.

Significant Figures Rules

The rules of significant figures are essential guidelines for determining which digits in a measurement are meaningful. Here's a simplified rundown:

• Non-zero digits are always significant.
• Any zeros between non-zero digits are significant.
• Leading zeros (zeros before all non-zero numbers) are not significant because they merely indicate position of the decimal point.
• Trailing zeros in a decimal number are significant as they indicate measured precision.
• In a whole number with no decimal point indicated, trailing zeros may or may not be significant depending on how the number was measured or reported.

Using these rules, one can determine the level of precision in a measurement and ensure that calculations are done without overestimating the accuracy. It's like a language that helps scientists and engineers communicate their findings with a clear understanding of uncertainties involved. By the careful application of these rules, as shown in the step-by-step exercise solutions, students can confidently solve problems while preserving the integrity of the measurements.