Question

Use the following exact conversion factors to perform the stated calculations: $$\begin{array}{rl}5\frac{1}{2}\text{ yards }& =1\text{ rod }\\ 40\text{ rods }& =1\text{ furlong }\\ 8\text{ furlongs }& =1\mathrm{mile}\end{array}$$ a. The Kentucky Derby race is $1.25$ miles. How long is the race in rods, furlongs, meters, and kilometers? b. A marathon race is 26 miles, 385 yards. What is this distance in rods, furlongs, meters, and kilometers?

Short answer

The Kentucky Derby race is 10 furlongs, 400 rods, 2011.68 meters, and 2.01168 kilometers long. The marathon race is 208 furlongs, 70 rods, 42194.888 meters, and 42.194888 kilometers long.

Step-by-step solution

a. Convert 1.25 miles to rods and furlongs

First, we'll convert the given distance in miles into rods and furlongs using the given conversion factors. 1.25 miles = 1.25 x (8 furlongs/mile) = 10 furlongs 1 furlong = 40 rods (given) So, 10 furlongs = 10 x 40 rods = 400 rods

a. Convert 1.25 miles to meters and kilometers

Now we'll convert the given distance in miles into meters and kilometers. 1.25 miles = 1.25 x (1609.344 meters/mile) = 2011.68 meters Now, we'll convert meters to kilometers using the fact that 1 km = 1000 m: 2011.68 meters = 2011.68/1000 kilometers ≈ 2.01168 kilometers Hence, The Kentucky Derby race is 10 furlongs, 400 rods, 2011.68 meters, and 2.01168 kilometers long.

b. Convert 26 miles and 385 yards to rods and furlongs

We are given the marathon race distance as 26 miles and 385 yards. First, let's convert the given distance in miles and yards into rods and furlongs. 26 miles = 26 x (8 furlongs/mile) = 208 furlongs The given distance also has a fraction in yards. Let's convert 385 yards to rods. 5.5 yards = 1 rod (given) So, 385 yards = 385/5.5 rods ≈ 70 rods Now, we have the distance in furlongs and rods.

b. Convert 26 miles, 385 yards to meters and kilometers

Now we'll convert the given distance in miles and yards into meters and kilometers. 26 miles = 26 x (1609.344 meters/mile) = 41842.944 meters 385 yards = 385 x (0.9144 meters/yard) ≈ 351.944 meters Now, adding the distances to find the total distance in meters: 41842.944 meters + 351.944 meters = 42194.888 meters Now, we'll convert meters to kilometers using the fact that 1 km = 1000 m: 42194.888 meters = 42194.888/1000 kilometers ≈ 42.194888 kilometers Hence, The marathon race is 208 furlongs, 70 rods, 42194.888 meters, and 42.194888 kilometers long.

Key concepts

Conversion Factors

Conversion factors are essential tools in chemistry that allow scientists and students to convert units from one system to another. They are akin to a mathematical 'bridge' that helps in translating different units of measurement. For example, in an exercise where you're asked to convert miles to meters or kilometers, you will use the conversion factor that defines how many meters or kilometers are in one mile.
To determine which conversion factor to use, it's important to know both the starting unit and the unit you're converting to. In the mentioned example, the conversion factors used are grounded in the relations: 1 mile equals 1609.344 meters and 1 kilometer equals 1000 meters. By multiplying the distance in miles by the conversion factor, you obtain the result in meters or kilometers. It's vital to ensure the units cancel out correctly so that you end with the desired units. Moreover, when dealing with fractions such as yards to rods, appropriate conversion factors based on the given problem are applied.

• Identify the given unit and what unit you need to convert it to.
• Choose the correct conversion factor to use.
• Multiply the given value by the conversion factor, ensuring that units cancel out properly to get the desired unit.

Fractions in Conversion

It's worth noting that when conversions involve fractions, as in the conversion of yards to rods, one must be cautious to accurately handle the fraction part. This will involve basic fraction arithmetic or the use of a calculator for precision.

Distance Conversion

Distance conversion is the process of converting a measurement from one unit of distance to another using appropriate conversion factors. In the field of chemistry, accurate measurement of distance is not as common; however, understanding the process of conversion is fundamental in the broader scientific context.
Conversion from one unit to another is crucial when the given unit is not in the desired unit for a particular calculation or comparison. For instance, in the exercise provided, distances in miles had to be converted into rods, furlongs, meters, and kilometers. This required a series of conversion steps, ensuring accuracy at each stage.

• Determine the measurement and the units you are starting with (e.g., miles).
• Identify the units you need to convert to (e.g., furlongs, rods).
• Use the chain of conversions to go from the starting unit to the desired unit, step by step.
• Be mindful of additional details such as extra distances in yards that need to be included in the total distance.

Conversion in stages can sometimes be necessary, as direct conversion factors might not always be available. In such cases, converting to an intermediate unit and then to the desired unit ensures that you can always find a pathway to the correct measurement.

Metric System

The metric system is an international system of units (SI), which is based on the powers of ten, making it very easy to convert from one unit to another. Scientists, including chemists, use the metric system because of its worldwide standardization and ease of scale. For example, in the metric system, distances are often measured in meters, with larger and smaller distances represented by adding prefixes to indicate multiples of ten.
When you work with the metric system, you frequently use units like meters and kilometers for distance. As shown in the exercise, meters were converted to kilometers by dividing by 1000, reflecting the metric system's simplicity when dealing with large or small numbers.

• The basic unit of length in the metric system is the meter (m).
• Other units like kilometers (km), centimeters (cm), and millimeters (mm) are derived by adding prefixes to indicate multiples or fractions of the meter.
• Conversion between these units simply involves moving the decimal point since the system is based on powers of ten.

To convert within the metric system, no complex factors are usually required; it's often a matter of shifting the decimal point according to the required scale, a technique that is relatively straightforward compared to navigating between different unit systems.