Problem 51 - College Physics

Question

Express the following calculations using the proper number of significant figures: (a) \(12.634+2.1\), (b) \(13.5-2.134\) (c) \(\pi(0.25 \mathrm{~m})^{2}\) (d) \(\sqrt{2.37 / 3.5}\)

Step-by-Step Solution

Answer

(a) 14.7, (b) 11.4, (c) 0.20 m², (d) 0.82

Step 1: Addition

To perform the addition for part (a): 12.634 + 2.1. The number with the least number of decimal places (2.1) has one decimal place. Therefore, the result should also have one decimal place. Perform the addition: \[12.634 + 2.1 = 14.734\]Until one decimal place: \[14.734 \approx 14.7\]

Step 2: Subtraction

For the subtraction in part (b): 13.5 - 2.134. The number with the least decimal places is 13.5 with one decimal place. Therefore, the result should have one decimal place. Perform the subtraction:\[13.5 - 2.134 = 11.366\]Until one decimal place: \[11.366 \approx 11.4\]

Step 3: Area using Pi

For part (c): \(\pi(0.25 \, \text{m})^2\). The radius (0.25 m) has two significant figures, so the answer should also have two significant figures. Calculate the area:\[\pi \times (0.25)^2 = 0.19635...\]Rounding to two significant figures: \[0.19635 \approx 0.20\]

Step 4: Square Root Calculation

For part (d): \(\sqrt{2.37 / 3.5}\). First perform the division and then take the square root. The division: \[\frac{2.37}{3.5} = 0.677142857\]The number of significant figures follows the number with the least significant figures which is 3.5 (two significant figures).After finding the square root:\[\sqrt{0.677142857} \approx 0.82256\]Rounding to two significant figures: \[0.82256 \approx 0.82\]

Key Concepts

Addition and Significant Figures

Addition in math is simply combining two numbers to get their total sum. However, when performing addition with significant figures, it's important to pay attention to the least precise measurement. If you have numbers like 12.634 and 2.1, consider which number has fewer decimal places. Here, 2.1 has only one decimal place, so the final answer must reflect this with just one decimal place too.

• Add the numbers: 12.634 + 2.1 = 14.734
• Round it to the proper decimal place: 14.734 becomes 14.7, ensuring correct significant figures.

By following this rule, you ensure that your answer is as precise as the least precise measurement.

Subtraction and Significant Figures

Subtraction works similarly to addition when it comes to maintaining significant figures. You must consider the number with the fewest decimal places and limit your result to that number of decimal places. For example, if you are subtracting 2.134 from 13.5, you'll notice 13.5 has only one decimal place. Thus, your final answer should also have one decimal place.

• Perform the subtraction: 13.5 - 2.134 = 11.366
• Round the result: 11.366 becomes 11.4, ensuring it aligns with the significant figure rules.

By doing so, your calculations remain consistent with the precision of the input numbers.

Area Calculation with Significant Figures

When calculating the area of a circle using Pi ( π ), it's essential to ensure your result reflects the proper number of significant figures present in your measurements. The formula for the area of a circle is πr^2 , where r is the radius. If the radius is given as 0.25 meters, which has two significant figures, your area result should also maintain this level of precision.

• Calculate the area: π imes (0.25)^2 = 0.19635...
• Round to two significant figures: 0.19635 becomes 0.20

This practice ensures your answer is consistent with the measure of precision in your inputs.

Square Root Calculation and Significant Figures

Square root calculations, much like other mathematical operations, require attention to the significant figures of the given data. When you have an operation involving division and square roots, track the significant figures from the division step.For example, if calculating \(\sqrt{\frac{2.37}{3.5}}\):

• The division step gives: \(\frac{2.37}{3.5} = 0.677142857\)
• Since 3.5 has two significant figures, follow this rule throughout.
• Take the square root: \(\sqrt{0.677142857} \approx 0.82256\)
• Round to two significant figures: 0.82256 becomes 0.82

Consistently applying the rules of significant figures maintains the precision of your mathematical results.