Problem 11 - Prealgebra

Question

Convert each rate using dimensional analysis. $$20 \mathrm{mi} / \mathrm{h}=\square{ft} / \mathrm{min}$$

Step-by-Step Solution

Answer

20 mi/h is 1760 ft/min.

Step 1: Identify What to Convert

The problem asks us to convert a speed from miles per hour (mi/h) to feet per minute (ft/min).

Step 2: Convert Miles to Feet

First, we need to convert miles to feet. We know that 1 mile equals 5280 feet. Thus, 20 miles can be converted to feet by multiplying:\[ 20 \text{ mi} \times 5280 \text{ ft/mi} = 105600 \text{ ft} \]

Step 3: Convert Hours to Minutes

Next, convert the time unit from hours to minutes. Since 1 hour is equivalent to 60 minutes, the conversion for 1 hour is:\[ 1 \text{ h} \times 60 \text{ min/h} = 60 \text{ min} \]

Step 4: Calculate the Conversion to Feet per Minute

Now that both conversions are set, calculate the speed in feet per minute by dividing the total feet by the total minutes:\[ \frac{105600 \text{ ft}}{60 \text{ min}} = 1760 \text{ ft/min} \]

Step 5: Conclusion

The conversion from 20 miles per hour to feet per minute results in 1760 feet per minute.

Key Concepts

Unit Conversion

Unit conversion is a crucial skill in math and science. It involves changing a quantity from one unit to another to make calculations or comparisons more manageable.

• Every unit conversion requires a clear understanding of equivalent values. For example, in this exercise, we're converting from miles to feet, knowing that 1 mile equals 5280 feet.
• You multiply the original quantity by a conversion factor, which is a fraction that equals 1. This doesn't change the measurement's value, only its units.

Using conversion factors, you can switch between units without altering the quantity's inherent meaning. This might seem simple, but it forms the foundation for more complex calculations in many scientific fields. Isn't it fascinating how a single conversion can help us understand distances in different contexts?

Speed Conversion

Speed conversion is a type of unit conversion specifically dealing with rates of motion. Here, we're converting a speed expressed in miles per hour to feet per minute.

• First, convert the distance unit from miles to feet. This tells us how far an object travels in feet rather than miles.
• Next, change the time measurement from hours to minutes. This lets us understand the rate of motion in shorter, more precise time frames.

Both conversions must be accurate to achieve a reliable speed in the new units. It's essential for tasks like designing vehicles or optimizing transportation routes, where understanding speeds in different units can provide clearer insights.

Rate Calculation

Rate calculation helps us determine how one quantity changes concerning another. When converting speed, we're actually calculating a rate: distance over time.

• First, find the total distance and time using conversion factors. Here, 20 miles becomes 105600 feet, and 1 hour turns into 60 minutes.
• Then, divide the distance by time. This gives the speed in a new unit, measured as feet per minute instead of miles per hour.

This concept simplifies understanding motion and flow, whether tracking a car's speed or water flowing through a pipe. Rate calculations are an essential part of everyday problem-solving scenarios, helping us better plan and execute tasks.

Prealgebra

Prealgebra serves as the foundation for mastering more advanced math topics, including concepts like unit and speed conversion. In prealgebra, we

• learn to manipulate numbers and solve equations, which are crucial skills for dimensional analysis.
• develop our understanding of proportional relationships by working through various exercises, such as converting rates.

These foundational skills ensure that when faced with real-world problems, such as speed conversions, students have the tools and confidence to tackle them effectively. Building a solid prealgebra foundation enables the application of math to practical situations and lays the groundwork for higher mathematical learning.