Question

Multiply the following measurements and round off the answer: (a) \(1.25 \mathrm{~cm} \times 0.5 \mathrm{~cm}\) (b) \(2.55 \mathrm{~cm} \times 1.1 \mathrm{~cm}\) (c) \(12.0 \mathrm{~cm}^{2} \times 1.00 \mathrm{~cm}\) (d) \(22.1 \mathrm{~cm}^{2} \times 0.75 \mathrm{~cm}\)

Short answer

(a) 0.6 cm², (b) 2.8 cm², (c) 12.0 cm³, (d) 17 cm³.

Step-by-step solution

Multiplying the Measurements for Part (a)

Calculate the product of the measurements given for part (a): Multiply the values 1.25 cm and 0.5 cm. This results in \(1.25 \times 0.5 = 0.625\text{ cm}^2\). Since the least number of significant figures in the measurements is 1 (in 0.5), the result should be rounded to 1 significant figure: \(0.6\text{ cm}^2\).

Multiplying the Measurements for Part (b)

Calculate the product of the measurements given for part (b): Multiply the values 2.55 cm and 1.1 cm. This results in \(2.55 \times 1.1 = 2.805\text{ cm}^2\). The smaller number of significant figures is 2 (in 1.1), so round to 2 significant figures: \(2.8\text{ cm}^2\).

Multiplying the Measurements for Part (c)

Calculate the product of the measurements given for part (c): Multiply the values 12.0 cm² and 1.00 cm. This results in \(12.0 \times 1.00 = 12.0\text{ cm}^3\). The least number of significant figures is 3, so the result should be expressed with 3 significant figures: \(12.0\text{ cm}^3\).

Multiplying the Measurements for Part (d)

Calculate the product of the measurements given for part (d): Multiply the values 22.1 cm² and 0.75 cm. This results in \(22.1 \times 0.75 = 16.575\text{ cm}^3\). The smaller number of significant figures is 2 (in 0.75), so round to 2 significant figures: \(17\text{ cm}^3\).

Key concepts

Rounding Off

Rounding off numbers is an essential skill in mathematics and science, especially when dealing with measurements. It's used to simplify numbers without significantly altering their value. This becomes particularly important in contexts where we follow rules of significant figures, a concept used to represent the precision of measurements.
Significant figures are all the digits in a number that contribute to its accuracy, including all non-zero digits, any zeros between them, and any trailing zeros in a decimal. For example, in the number 1.25, all three digits are considered significant.

• When rounding, we look at the digit following the last desired significant figure. If it is 5 or greater, we round up.
• If it is less than 5, we round down.

In scientific calculations, it is essential to consider the significant figures of the numbers involved. For example, when multiplying 1.25 cm by 0.5 cm in part (a) of the problem, although the mathematical result is 0.625, it needs to be expressed with the correct number of significant figures—in this case, just one, because 0.5 has only one significant figure. Thus, it rounds to 0.6.

Measurement Multiplication

Measurement multiplication deals with calculating the product of different quantities while respecting the rules of significant figures. Multiplying measurements involves straightforward arithmetic, but you need to apply the significant figures rule to the final answer to maintain accuracy.
For example, in part (b) of the original exercise, multiplying 2.55 cm by 1.1 cm equals 2.805 cm² mathematically. However, the rule dictates that the product should have the same number of significant figures as the multiplier with the least. Here, 1.1 cm has two significant figures, so the result should be rounded to two significant figures, yielding 2.8 cm².

• The rule ensures the precision of the result is not overstated beyond the original measurements.
• Accurate multiplication of measurements can often lead to changes in the units, like squaring or cubing them, as seen in the exercise, where cm becomes cm² or cm³.

Units of Measurement

Understanding and using the correct units of measurement is fundamental when performing mathematical operations involving physical quantities. Units provide a standard measure that quantifies the physical quantity, and it is crucial to keep track of them during calculations to ensure the result remains meaningful.
When measurements are multiplied, their units are also multiplied. For example, multiplying cm by cm results in cm², representing area, while multiplying cm² by cm gives cm³, which represents volume. These transformations reflect the dimensional nature of the physical quantities involved, like in part (c), where two-dimensional area (cm²) is multiplied by another measurement (cm) resulting in three-dimensional volume (cm³).

• Units help communicate what a measurement represents, ensuring clarity and avoiding misunderstanding.
• Always include units in calculations to maintain physical relevance and accuracy.

In science and engineering, proper use of units can also point to mistakes if the final unit does not match the expected physical quantity.