Ch. 1 Worksheets Answer Key.pdf - Chemistry IF8766 ANSWER KEY ... - Free Printable

Problem Analysis:

The task involves solving a problem using dimensional analysis (also known as the factor-label method). The goal is to convert units step by step, ensuring that the units cancel out appropriately to yield the desired final unit.

#### Given Information:
- Initial Quantity: 1.00 mile
- Final Unit: centimeters (cm)
- Conversion Factors:
1. 1 mile = 5280 feet
2. 1 foot = 12 inches
3. 1 inch = 2.54 cm

#### Objective:
Convert 1.00 mile into centimeters using dimensional analysis.

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Step-by-Step Solution:

#### Step 1: Write Down the Initial Quantity
Start with the given quantity:
$$
1.00 \text{ mile}
$$

#### Step 2: Apply the First Conversion Factor
Use the conversion factor \( 1 \text{ mile} = 5280 \text{ feet} \):
$$
1.00 \text{ mile} \times \frac{5280 \text{ feet}}{1 \text{ mile}}
$$
The "miles" unit cancels out:
$$
1.00 \cancel{\text{ mile}} \times \frac{5280 \text{ feet}}{1 \cancel{\text{ mile}}} = 5280 \text{ feet}
$$

#### Step 3: Apply the Second Conversion Factor
Use the conversion factor \( 1 \text{ foot} = 12 \text{ inches} \):
$$
5280 \text{ feet} \times \frac{12 \text{ inches}}{1 \text{ foot}}
$$
The "feet" unit cancels out:
$$
5280 \cancel{\text{ feet}} \times \frac{12 \text{ inches}}{1 \cancel{\text{ foot}}} = 63360 \text{ inches}
$$

#### Step 4: Apply the Third Conversion Factor
Use the conversion factor \( 1 \text{ inch} = 2.54 \text{ cm} \):
$$
63360 \text{ inches} \times \frac{2.54 \text{ cm}}{1 \text{ inch}}
$$
The "inches" unit cancels out:
$$
63360 \cancel{\text{ inches}} \times \frac{2.54 \text{ cm}}{1 \cancel{\text{ inch}}} = 160934.4 \text{ cm}
$$

#### Step 5: Round to Appropriate Significant Figures
The initial value \( 1.00 \text{ mile} \) has three significant figures. Therefore, the final answer should also have three significant figures. Rounding \( 160934.4 \text{ cm} \) to three significant figures gives:
$$
161000 \text{ cm}
$$

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Final Answer:

$$
\boxed{161000 \text{ cm}}
$$

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Explanation of the Process:

1. Dimensional Analysis: This method ensures that units are treated algebraically, allowing them to cancel out in a chain of conversions.
2. Conversion Factors: Each step uses a conversion factor that relates the current unit to the next desired unit. The numerator and denominator of each fraction are chosen so that unwanted units cancel out.
3. Significant Figures: The final answer is rounded to match the precision of the given data (three significant figures in this case).

This approach is systematic and minimizes errors by keeping track of units throughout the calculation.