Problem 79 - Introductory Chemistry Atoms First

Question

A student walks \(20,450.2 \mathrm{ft}\) to school every day. A mile is defined as \(5280 \mathrm{ft}\). Doing the division \(20,450.2 \mathrm{ft} \div 5280 \mathrm{ft}\) per mile on a calculator gives \(3.873143939\) miles. What, if anything, is wrong with this answer?

Step-by-Step Solution

Answer

The given answer is correct. The student walks approximately \(3.873143939 \textrm{ miles}\) to school every day.

Step 1: Identify the Conversion Factor

We are given that \(1 \textrm{ mile } = 5280\mathrm{ ft}\). This conversion factor will be used to find the distance in miles.

Step 2: Convert the Distance in Feet to Miles

Given the student walks \(20,450.2 \mathrm{ ft}\) to school every day, we can find the distance in miles by dividing the distance in feet by the conversion factor. To do this, we divide \(20,450.2 \mathrm{ft}\) by the number of feet in a mile (\(5280 \mathrm{ft}\)): \[ \frac{20,450.2 \mathrm{ ft}}{5280 \mathrm{ ft/mile}} \]

Step 3: Perform the Division and Check the Answer

Now perform the division: \[ \frac{20,450.2}{5280} = 3.873143939 \] Comparing this value to the given answer, we find that they match: \[ 3.873143939 \textrm{ miles} \] There doesn't seem to be anything wrong with the given answer. Therefore, the student walks approximately \(3.873143939 \textrm{ miles}\) to school every day.

Key Concepts

Dimensional Analysis

Dimensional analysis is a mathematical technique used in chemistry and other sciences to convert one unit of measurement to another. Its importance can't be overstressed, as it ensures that various measurements can be accurately compared and utilized — critical not just in academic exercises, but in real-world applications such as calculating dosages in medicine or converting currency for international transactions.

In the exercise provided, the student needed to convert feet to miles. Dimensional analysis comes into play by using the conversion factor that relates these two units. This involves setting up a ratio that allows for the cancellation of units, and ensuring that the final calculation retains the correct dimension — in this case, miles. During this process, it's essential to arrange the conversion factor so that the unwanted unit (in this case, feet) cancels out to leave the desired unit (miles).

In educational terms, it’s helpful for students to visualize unit conversion as a bridge that they cross to translate one unit into the terms of another while keeping the physical quantity consistent. Understanding and performing dimensional analysis cultivates precision in scientific calculations and reinforces a student's comprehension of measurement concepts.

Conversion Factors

Conversion factors are used to change one unit of measure to another, enabling us to perform calculations involving different units. They are absolutely essential in dimensional analysis. A conversion factor is a ratio that expresses how many of one unit equals another unit. It's the tool we use to 'convert' quantities into different units of measure.

Considering the given exercise, the conversion factor is that 1 mile equals 5280 feet. This factor is crucial because it's the bridge that connects feet to miles, permitting the translation of one measure to the other seamlessly. When using conversion factors, it's crucial for students to understand that they are equivalent values; thus, multiplying by the conversion factor is akin to multiplying by one, which doesn't change the quantity, only the units.

Using the Correct Conversion Factor

Selecting the correct conversion factor, as seen in the problem's step 1, is a critical thinking step that involves understanding the relationship between the units involved and conceptualizing the direction of the conversion.

Significant Figures

When dealing with measurements and conversions in chemistry, we cannot ignore the significant figures. Significant figures, commonly referred to as 'sig figs,' represent all the known digits in a number plus the first uncertain digit. They are intrinsic to maintaining precision in scientific calculations.

In the original exercise, the calculator gave the answer as 3.873143939 miles when we converted feet to miles. However, we must consider significant figures when reporting our final answer. If we look at the lengths given, we notice that the largest number of decimal places is one, as in 20,450.2 feet. Therefore, our answer shouldn't be more precise than the least precise measurement we started with.

Applying Significant Figures in the Solution

In the student's calculation, the answer should be rounded to the nearest tenth of a mile considering the original data's precision. It means the answer should be reported as approximately 3.9 miles, not the longer, overly precise value provided by the calculator. This rounding reflects the limit of precision in the original measurement and demonstrates an understanding of measurement uncertainty in scientific experiments.