Question

You are asked to determine the perimeter of the cover of your textbook. You measure the length as \(34.29 \mathrm{~cm}\) and the width as 26.72 \(\mathrm{cm} .\) How many significant figures should you report for the perimeter?

Short answer

The perimeter of the textbook cover should be reported as 122.0 cm with four significant figures.

Step-by-step solution

Identify the given measurements and their significant figures

The length of the textbook cover is 34.29 cm and the width is 26.72 cm. Both measurements have four significant figures.

Calculate the perimeter of the rectangle

To find the perimeter of a rectangle, we use the formula \(P = 2l + 2w\), where \(P\) is the perimeter, \(l\) is the length, and \(w\) is the width. Substitute the given values into the equation: \(P = 2(34.29 \mathrm{~cm}) + 2(26.72 \mathrm{~cm})\)

Perform the calculations

Calculate the value of the perimeter: \(P = 68.58 \mathrm{~cm} + 53.44 \mathrm{~cm} = 122.02 \mathrm{~cm}\)

Determine the number of significant figures in the final answer

Since both given measurements have four significant figures and we were adding or subtracting them, the final answer should also have four significant figures. Therefore, the perimeter should be reported as: P = 122.0 cm

Key concepts

Perimeter Calculation

Understanding how to calculate the perimeter of an object is a fundamental skill in geometry. The perimeter is the total distance around the outermost edges of a two-dimensional shape. For rectangles and squares, this is quite straightforward: you simply add up the lengths of all four sides. In the context of our exercise, the perimeter (\( P \) of a rectangle is given by the formula \( P = 2l + 2w \), where \( l \) represents the length and \( w \) represents the width of the rectangle. If a textbook cover measures \( 34.29 \) cm by \( 26.72 \) cm, we double each measurement (to account for the two lengths and two widths) and add them to find the total perimeter.

Scientific Measurement

In scientific measurement, precision and accuracy are paramount. Each measurement comes with a degree of uncertainty, typically determined by the instrument used to make the measurement. Significant figures are a way of expressing this accuracy. They include all certain digits in a measurement plus the first uncertain digit. For instance, when you measure the length of an object as \( 34.29 \) cm, you are expressing that you are certain about the '34', '2', and '9', with '9' being the least certain digit. Measurements are often rounded to the appropriate number of significant figures to reflect the precision of the measurement tool. In science and engineering, proper reporting of significant figures is crucial to ensure that calculations are accurate and not overrepresented.

Addition and Subtraction of Significant Figures

When dealing with the addition or subtraction of measurements, the rule for significant figures is to look at the decimal places. The result should be reported with the same number of decimal places as the measurement with the fewest decimal places. This is because adding or subtracting numbers affects the precision of the result to that of the least precise measurement. In our textbook cover example, after calculating the raw perimeter as \( 122.02 \) cm, we notice that each original measurement had two decimal places. Therefore, despite the raw sum having more digits, the final perimeter should be rounded to \( 122.0 \) cm to properly reflect the precision of our most uncertain measurement. This aligns the perimeter's significant figures with the significant figures in the original measurements.