Question

A poster is \(0.95 \mathrm{~m}\) high and \(1.0 \mathrm{~m}\) wide. How many digits follow the decimal point when the perimeter of the poster is expressed with the correct number of significant figures?

Short answer

The perimeter has 1 digit following the decimal point, matching the significant figures.

Step-by-step solution

Identify the Given Dimensions

We are given the dimensions of the poster: height = \(0.95 \mathrm{~m}\) and width = \(1.0 \mathrm{~m}\). Each measurement has its own precision based on the number of significant figures.

Calculate the Perimeter

The formula for the perimeter \( P \) of a rectangle is \( P = 2 imes ( ext{height} + ext{width}) \). Substitute the given values: \( P = 2 imes (0.95 + 1.0) = 2 imes 1.95 = 3.9 \mathrm{~m}\).

Determine the Number of Significant Figures

The height \(0.95\) has two significant figures, and the width \(1.0\) also has two significant figures. The result of a calculation should be expressed with the least number of significant figures from the given values, which is two.

Check the Decimal Places in the Perimeter Result

The calculation results in a perimeter of \(3.9\), which already has two significant figures. The number of digits after the decimal point in this result is 1.

Key concepts

Measurement Precision

Measurement precision refers to how accurate a measurement is, depending on how detailed or specific the measurement instrument is. It is a reflection of the smallest increment or value that can be measured with certainty. For instance, if you measure a length with a ruler marked in millimeters, your measurement is more precise than if you were using a ruler marked in centimeters.
Precision is crucial when performing calculations to ensure results are reliable and consistent. The measurements given for the poster, such as height at \(0.95\) meters and width at \(1.0\) meters, come with a certain precision, reflected in the number of decimal places and significant figures used. When doing calculations or making comparisons, it's essential to consider this precision.
In the poster example, we know these measurements are precise to the hundredths (two decimal places) and tenths (one decimal place), respectively, which impacts how we report and interpret the final calculation.

Significant Digits

Significant digits are the digits in a number that contribute meaning or value to the measurement. They include all non-zero digits, any zeros between significant digits, and any trailing zeros in a decimal number. Understanding significant digits is fundamental in reporting scientific or measured data accurately.
When performing calculations, such as calculating the perimeter of the poster, it's crucial to report the result using the correct number of significant digits. This ensures that the result is neither over- nor under-estimated based on the precision of the input measurements.
For the poster, both height and width are reported using two significant digits. Thus, the perimeter calculation must also be expressed with two significant figures, which explains why the final result of the perimeter is given as \(3.9\) meters. By maintaining this rule, we maintain the integrity of the data and the precision of the measurements involved.

Perimeter Calculation

Perimeter calculation is a simple yet fundamental arithmetic operation used to find the total distance around a shape. For rectangles, like the poster in question, the formula is straightforward: \(P = 2 \times (\text{height} + \text{width})\). This formula efficiently accounts for all sides of the rectangle.
In practice, applying this to the poster dimensions given, we plug in the values: \(P = 2 \times (0.95 \text{ m} + 1.0 \text{ m})\). This results in \(2 \times 1.95\), yielding \(3.9 \text{ m}\). Such calculations are routine yet essential for ensuring the correct and consistent application of mathematical principles in measurements.
This calculation helps illustrate the use of significant figures in practical applications, highlighting the importance of mathematics in accurately determining dimensions and ensuring precise measurements are used in all aspects of design and construction.

Decimal Place Accuracy

Decimal place accuracy signifies how detailed a number is in terms of digits following the decimal point. It is important when expressing measurements because it communicates the degree of precision. More decimal places generally indicate greater accuracy.
In the context of the perimeter calculation for the poster, the result is \(3.9\) meters. This number has one digit after the decimal point, which reflects both the precision of the original measurements and the significant figures involved.
By ensuring our perimeter calculation respects the decimal place accuracy of input values, we're able to provide a realistic and high-confidence figure. Understanding decimal place accuracy helps prevent errors in reporting and ensures that mathematical results are trusted and valid in practical scenarios.