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Chapter 2: Problem 9

Which is longer: a 1 -mile race or a 1500 -meter race? Show your reasoning.

Short Answer

Expert verified
A 1-mile race is longer than a 1500-meter race.

Step by step solution

01

Understand the Conversion

To compare a 1-mile race and a 1500-meter race, we need to have both distances in the same unit of measurement.
02

Convert Miles to Meters

1 mile is equivalent to approximately 1609.34 meters. This conversion will allow us to directly compare the two races.
03

Compare the Distances

Now that we know 1 mile is 1609.34 meters, we can compare it to 1500 meters. Since 1609.34 meters is greater than 1500 meters, the 1-mile race is longer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Metric System
The metric system is a decimal-based system of measurement used widely around the world. It's preferred for its simplicity and ease of use. The standard unit of length in the metric system is the meter. Other units include kilometers, centimeters, and millimeters. These units are proportional to each other in factors of ten, making conversion straightforward.
For example:
  • 1 kilometer = 1000 meters
  • 1 meter = 100 centimeters
  • 1 centimeter = 10 millimeters
This helps when visually interpreting distances and performing calculations, as you only need to move the decimal point to convert between units.
Imperial System
The imperial system is used primarily in the United States and consists of units like inches, feet, yards, and miles. Unlike the metric system, conversions between units aren't based on a base-10 system, which can sometimes make it more complicated. For instance, 1 mile is equal to 5280 feet, which does not align neatly with metric conversions.
Key length conversions in the imperial system include:
  • 1 foot = 12 inches
  • 1 yard = 3 feet
  • 1 mile = 1760 yards
Understanding these conversions is crucial when comparing measurements from the imperial system with those from the metric system, as seen in races and other sports events.
Mathematical Reasoning
Mathematical reasoning is the ability to think logically about problems and utilize mathematical concepts to solve them. In the given exercise, we use reasoning to determine which race is longer by converting different units into the same metric.
Steps of reasoning include:
  • Identifying the need for conversion to compare values directly
  • Finding the correct conversion factor
  • Applying this factor to reach a conclusion
This type of reasoning allows us to make informed decisions and comparisons, crucial not only in academics but also in daily life scenarios.
Unit Conversion
Unit conversion is an essential mathematical process that involves changing a measurement from one unit to another. In the race comparison problem, it is required to convert miles to meters or vice versa to make an accurate comparison.
Here's how to perform a unit conversion:
  • Determine the unit you are starting with and the unit you need to convert to.
  • Use the appropriate conversion factor. For example, 1 mile is 1609.34 meters.
  • Multiply the original measurement by the conversion factor to get the equivalent measurement in the desired unit.
Understanding how to convert units accurately ensures precision in sciences, engineering, and many real-world applications.

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Most popular questions from this chapter

This problem is sometimes called Einstein's problem: "Use the digits \(1,2,3,4,5,6,7,8,9\) and any combination of the operation signs \((+,-,+,\), to write an expression that equals 100 . Keep the numbers in consecutive order and do not use parentheses." Here is one solution: $$ 123-4-5-6-7+8-9=100 $$ Your task is to find another one.

A bug is crawling horizontally along the wall at a constant rate of 5 inches per minute. You first notice the bug when it is in the corner of the room, behind your music stand. a. Define variables and write an equation that relates time (in minutes) to distance traveled (in inches). (a) b. What is the constant of variation of this direct variation relationship, and what does it represent? c. How far will the bug crawl in \(1 \mathrm{~h}\) ? d. How long would you have to practice playing your instrument before the bug completely "circled" the \(14-\mathrm{ft}-\mathrm{by}-20\)-ft room? (a) e. Draw a graph that represents this situation.

As part of their homework assignment, Thu and Sabrina each found equations from a table of data relating miles and kilometers. One entry in the table paired 150 kilometers and 93 miles. From this pair of data values, Thu and Sabrina wrote different equations. a. Thu wrote the equation \(y=1.61 x\). How did he get it? What does \(1.61\) represent? What do \(x\) and \(y\) represent? (Ii) b. Sabrina wrote \(y=0.62 x\) as her equation. How did she get it? What does \(0.62\) represent? What do \(x\) and \(y\) represent? c. Whose equation would you use to convert miles to kilometers? d. When would you use the other student's equation?

Describe how to solve each equation for \(x\). Then solve. a. \(14=3.5 x\) (a b. \(8 x=45(0.62)\) c. \(\frac{x}{7}=0.375\) d. \(\frac{12}{x}=0.8\)

Evaluate each expression if \(x=6\). a. \(2 x+3\) b. \(2(x+3)\) c. \(5 x-13\) d. \(\frac{x+9}{3}\)
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