Problem 51 - Chemistry: The Central Science

Question

By using estimation techniques, arrange these items in order from shortest to longest: a \(57-\mathrm{cm}\) length of string, a 14 -in. long shoe, and a \(1.1-\mathrm{m}\) length of pipe.

Step-by-Step Solution

Answer

First, convert all lengths to centimeters: the 14-inch shoe is approximately 35.56 cm long, and the 1.1-meter pipe is 110 cm long. Then arrange the items from shortest to longest: shoe (35.56 cm), string (57 cm), pipe (110 cm).

Step 1: Convert all lengths to centimeters

Firstly, we need to convert the given lengths into a common unit, which is centimeters in this case. The length of the string is already in centimeters (57 cm). Now, we will convert the other lengths. To convert the 14-inch long shoe to centimeters, we can use the following formula: Length in centimeters = Length in inches * 2.54 To convert the 1.1-meter long pipe to centimeters, we can use the following formula: Length in centimeters = Length in meters * 100

Step 2: Calculate the length of the shoe in centimeters

Using the formula to convert inches to centimeters, we can calculate the length of the shoe: Length of the shoe in centimeters = 14 inches * 2.54 cm/inch ≈ 35.56 cm

Step 3: Calculate the length of the pipe in centimeters

Using the formula to convert meters to centimeters, we can calculate the length of the pipe: Length of the pipe in centimeters = 1.1 meters * 100 cm/meter = 110 cm

Step 4: Arrange the items in order from shortest to longest

Now that all items are in the same unit (centimeters), we can arrange them in order from shortest to longest: 1. Shoe: 35.56 cm 2. String: 57 cm 3. Pipe: 110 cm So, the order of these items from shortest to longest is the shoe, the string, and the pipe.

Key Concepts

Estimation Techniques

Estimation techniques are essential in many aspects of daily life and particularly useful in solving mathematical problems. They allow us to make educated guesses without requiring exact numbers, which can save time and effort. For example, when comparing the lengths of objects that are given in different units, like in our exercise with the string, shoe, and pipe, a rough comparison can quickly determine their order by size without precise measurement.

Let's apply estimation to our scenario. Understanding that 1 inch is approximately 2.5 centimeters (cm) can help us estimate that the 14-inch shoe is roughly 35cm (since 14 times 2 is 28, and a little more for the additional 0.5 per inch gives us about 35). Similarly, knowing that 1 meter is 100cm makes it clear that the 1.1-meter pipe is just over 100cm. Through estimation, the order is evidently the shoe, the string, and then the pipe, which aligns with our precise calculations.

Metric System

The metric system is an internationally adopted decimal system of measurement. It is preferred due to its ease of use and the fact that it is based on multiples of ten. Key units in the metric system include the meter for length, the kilogram for mass, and the second for time. For our exercise, we particularly care about the meter and its subunit the centimeter. One meter equals 100 centimeters, which is a simple multiplication factor.

Conversion within Metric Units

To convert from meters to centimeters, as required in our exercise, you multiply the number of meters by 100. Similarly, to convert back to meters from centimeters, you divide the number of centimeters by 100. This simplicity makes the metric system straightforward for performing unit conversions and comparisons, like converting the 1.1-meter pipe to 110 centimeters.

Imperial System

In contrast to the metric system, the imperial system is commonly used in the United States and a few other countries. It uses different units such as inches, feet, and yards for measuring lengths. The system is not based on a decimal base, making the conversions less intuitive.

Converting Imperial to Metric

When converting from imperial to metric, specific conversion factors are used. The exercise provided shows how to convert inches to centimeters using a conversion factor of 2.54 (1 inch is equal to 2.54 centimeters). To convert the 14-inch shoe, we multiply the number of inches by 2.54 to obtain the length in centimeters. These conversion steps are crucial for accurate and effective comparisons between units from different measurement systems.