Problem 45 - Prealgebra

Question

An airplane flew from Boston through Chicago to Denver. The distance from Boston to Chicago was 1015 miles, and the distance from Chicago to Denver was 1011 miles. The plane traveled for 3.5 hours and carried 285 passengers. About how fast did the airplane travel?

Step-by-Step Solution

Answer

The airplane traveled at approximately 579 miles per hour.

Step 1: Understand the Total Distance

First, calculate the total distance the airplane traveled by adding the distance from Boston to Chicago and the distance from Chicago to Denver. This is given by 1015 miles + 1011 miles.

Step 2: Calculate the Total Distance

Add the distances: \(1015 + 1011 = 2026\) miles. So, the airplane traveled a total of 2026 miles.

Step 3: Use the Formula for Average Speed

The formula to find the average speed is \(\text{Speed} = \frac{\text{Total Distance}}{\text{Total Time}}\). You were given the total time as 3.5 hours.

Step 4: Plug Values into the Formula

Substitute the total distance and total time into the formula: \(\text{Speed} = \frac{2026 \text{ miles}}{3.5 \text{ hours}}\).

Step 5: Calculate the Average Speed

Divide 2026 by 3.5 to find the average speed of the plane: \(\frac{2026}{3.5} \approx 578.86\) miles per hour.

Key Concepts

Distance Measurement

Understanding how to measure distance is a fundamental concept in solving many math problems, including average speed calculations. In this exercise, we were given two separate distances: 1015 miles from Boston to Chicago and 1011 miles from Chicago to Denver. Step by step, these distances were combined to determine the total distance traveled by the airplane.

• The key operation here is addition.
• When you add the two segments together, you get the total journey: 1015 miles + 1011 miles.
• This gives a total distance of 2026 miles.

Distance is usually measured in units such as miles or kilometers. It's important to be consistent with the unit you are using throughout your calculations. Here, the unit was miles, which is commonly used in the United States for measuring lengthy distances. This approach prepares the ground for calculating average speed and other related measures.

Time Calculation

Time calculation involves breaking down the duration of an event, which is crucial when determining average speed. In the problem, the airplane's total travel time of 3.5 hours was provided.

• Time is a continuous measure usually expressed in hours, minutes, and seconds.
• Understanding how to use decimals can make calculations more straightforward, such as interpreting 3.5 hours as 3 hours and 30 minutes.

Converting time to different units is often necessary, especially if times are given in mixed units. For example, converting milliseconds to seconds or minutes to hours may come into play in more complex problems. For this exercise, however, the time was already compatible with our needs for calculating average speed.

Mathematics Problem-Solving

Problem-solving in mathematics involves systematic approaches and selecting the correct formulas and operations. To solve the average speed of the airplane, a clear plan was followed.

• First, the total distance was identified and computed.
• Then, the total time was either given or calculated based on the problem.
• The formula for average speed, which is \(\text{Speed} = \frac{\text{Total Distance}}{\text{Total Time}}\), was then applied.

Implementing correct operations at each step and inputting the correct values into the formula was crucial. Mathematics problem-solving often requires you to interpret word problems into mathematical expressions, ensuring no step is skipped or misunderstood.

Prealgebra Concepts

Prealgebra serves as the foundation for approaching more advanced topics in mathematics, such as algebra or calculus. In this typical prealgebra exercise, the core idea revolves around understanding relationships between numbers and applying basic arithmetic operations.

• Key concepts include addition, division, and the understanding of mathematical formulas.
• Substituting numbers into equations and calculating using these numbers was essential.

Prealgebra concepts often focus on concrete examples, like this exercise about an airplane's journey, to ground abstract ideas in real-world situations. Grasping these basics helps set you on the right path for tackling increasingly complex mathematical challenges as you continue your education.