Problem 34 - Prealgebra

Question

Convert each rate using dimensional analysis. $$32 \mathrm{cm} / \mathrm{s}=\square \mathrm{m} / \mathrm{min}$$

Step-by-Step Solution

Answer

32 cm/s converts to 19.2 m/min.

Step 1: Identify the conversion factors

To convert from centimeters per second to meters per minute, identify the conversion factors needed: 100 cm = 1 m and 60 seconds = 1 minute.

Step 2: Convert centimeters to meters

Use the conversion factor for length to convert centimeters to meters: \[32 \text{ cm/s} \times \frac{1 \text{ m}}{100 \text{ cm}} = 0.32 \text{ m/s}.\]

Step 3: Convert seconds to minutes

Next, use the conversion factor for time to convert per second to per minute:\[0.32 \text{ m/s} \times 60 \text{ seconds/minute} = 19.2 \text{ m/min}.\]

Step 4: Write the final answer

After the conversions, the rate in meters per minute is:\[32 \text{ cm/s} = 19.2 \text{ m/min}.\]

Key Concepts

Unit Conversion

Unit conversion is a fundamental part of dimensional analysis. It's the process of converting a measurement from one unit to another, thus allowing for consistent measurements across different systems. In the context of the exercise, we converted a speed from centimeters per second (cm/s) to meters per minute (m/min). This requires keen attention to both the units you're converting from and the units you're converting to.

To successfully perform unit conversion, it's essential to understand and memorize common conversion factors. In the exercise, we used two key conversion factors:

• Length Conversion: 100 centimeters (cm) is equivalent to 1 meter (m).
• Time Conversion: 60 seconds (s) is equivalent to 1 minute (min).

Utilizing these conversion factors, you can accurately convert measurements from one unit to their equivalents in another unit. Remember, accurate unit conversion enables scientists, engineers, and everyday people to compare measurements and make calculations consistently.

Metric System

The metric system is a decimal-based system of measurement used worldwide, renowned for its simplicity and ease of use. It is based on powers of ten, making conversions straightforward—simply shift the decimal point to convert between units. In the context of our example, converting from centimeters to meters aligns perfectly with the metric system's principles.

Within the metric system, units are systematically organized:

• For length: millimeters (mm), centimeters (cm), meters (m), and kilometers (km).
• For volume: milliliters (mL) and liters (L).
• For mass: grams (g) and kilograms (kg).

Understanding these units and their relationships is crucial for effectively performing unit conversions. The metric system's logic and number-based hierarchy simplify tracking and converting between different scales of measurement. So next time you're faced with conversions, recall the elegant simplicity of the metric system to guide you through.

Rates and Ratios

Rates and ratios are expressions comparing two quantities. They often appear in measurements that cover time, such as speed or acceleration. In our exercise, the rate involves length per unit of time: centimeters per second (cm/s) translating to meters per minute (m/min). This concept requires careful attention to both parts of the rate—numerical value and its units.

Rates are essentially ratios representing relationships between two units. For example, rates like speed combine distance and time. When converting rates, each part of the ratio needs to be converted accordingly. Steps to ensure successful rate conversions include:

• Identify both the numerator and the denominator's units.
• Convert each unit appropriately using reliable conversion factors.
• Ensure that all parts of the rate are consistent with each other before concluding.

Whether solving a physics equation or determining travel speed, mastering the manipulation of rates and ratios ensures precise calculations and better understanding of relationships between various quantities.