Question

Multiply the following measurements and round off the answer: (a) \(3.65 \mathrm{~cm} \times 2.10 \mathrm{~cm}\) (b) \(8.75 \mathrm{~cm} \times 1.15 \mathrm{~cm}\) (c) \(16.5 \mathrm{~cm} \times 1.7 \mathrm{~cm}\) (d) \(21.1 \mathrm{~cm} \times 20 \mathrm{~cm}\)

Short answer

(a) 7.67 cm extsuperscript{2}, (b) 10.1 cm extsuperscript{2}, (c) 28 cm extsuperscript{2}, (d) 400 cm extsuperscript{2}.

Step-by-step solution

Understanding Significant Figures

Significant figures are the digits in a number that contribute to its precision. When multiplying, the result should be rounded to the same number of significant figures as the measurement with the fewest significant figures.

Calculate Product for (a)

Multiply the measurements for part (a): \[ 3.65 \text{ cm} \times 2.10 \text{ cm} = 7.665 \text{ cm}^2 \] The number with the fewest significant figures is 2.10 (3 significant figures). Thus, round 7.665 to 3 significant figures: \[ 7.67 \text{ cm}^2 \]

Calculate Product for (b)

Multiply the measurements for part (b): \[ 8.75 \text{ cm} \times 1.15 \text{ cm} = 10.0625 \text{ cm}^2 \] The number with the fewest significant figures is 1.15 (3 significant figures). Thus, round 10.0625 to 3 significant figures: \[ 10.1 \text{ cm}^2 \]

Calculate Product for (c)

Multiply the measurements for part (c): \[ 16.5 \text{ cm} \times 1.7 \text{ cm} = 28.05 \text{ cm}^2 \] The number with the fewest significant figures is 1.7 (2 significant figures). Thus, round 28.05 to 2 significant figures: \[ 28 \text{ cm}^2 \]

Calculate Product for (d)

Multiply the measurements for part (d): \[ 21.1 \text{ cm} \times 20 \text{ cm} = 422 \text{ cm}^2 \] The number with the fewest significant figures is 20 (1 significant figure, since only '2' is considered significant and it represents 20). Thus, round 422 to 1 significant figure: \[ 400 \text{ cm}^2 \]

Key concepts

Significant Figures and Rounding

Rounding is a method used to simplify numbers while maintaining their essential precision. When working with significant figures, rounding becomes crucial as it reflects the accuracy of your measurements. When dealing with multiplication or division of measurements, the number of significant figures in the result should match the number of significant figures in the measurement with the least number of significant figures. This ensures you do not overstate the precision of your result.

For instance, consider the problem step involving (a) where we multiplied 3.65 cm and 2.10 cm resulting in 7.665 cm². Since 2.10 cm contains the fewest significant figures (3), we round 7.665 to three significant figures, leading us to 7.67 cm². This approach ensures that the answer is as precise as the least precise measurement involved, which is a fundamental rule in scientific calculations.

Measurement Multiplication

When we multiply measurements, we multiply their numerical values and their units separately. The product of the values gives us a direct answer, but its precision relies heavily on the concept of significant figures.

In exercise (b), we carried out the multiplication of 8.75 cm by 1.15 cm and obtained 10.0625 cm². However, just obtaining a raw product isn’t sufficient. Since both measurements have three significant figures, our product must be rounded to three significant figures too, resulting in 10.1 cm². By adjusting the precision here, we reflect the limitation imposed by the input measurements and enhance the reliability of our result.

• Math operations with measurements need careful handling for precision.
• Results are tightened to match the least precise measurement.

Precision in Calculations

Precision in calculations is about maintaining the integrity and reliability of the derived results. In real-world measurements, no measurement is infinitely precise, and it is important to carry this lack of absolute precision through subsequent calculations by considering significant figures.

In step (c), multiplying 16.5 cm and 1.7 cm, we achieved a product of 28.05 cm². The limiting factor here was 1.7 cm, which possesses two significant figures, urging us to round our answer to just two significant figures, thus giving 28 cm².

Calculating with precision requires an awareness of where limitations exist and involves adhering closely to the measurement with the least precision. This ensures that when we combine, compare, or report values, they accurately reflect the original measurement conditions rather than misleading with an apparent, but unfounded, accuracy.