Problem 17 - Algebra 1

Question

In Exercises \(15-18\), find the average speed. Swim 2 miles in 40 minutes

Step-by-Step Solution

Answer

The average speed is 3 miles per hour.

Step 1: Convert time to hours

Convert the time from minutes to hours since speed is typically measured in miles per hour. To do this, divide the number of minutes by the number of minutes in an hour (60 minutes). Therefore, \( 40 \, minutes = \frac{40}{60} \, hours = \frac{2}{3} \, hours \).

Step 2: Calculate average speed

Average speed is calculated by dividing the distance traveled by the time taken. Here, the distance is 2 miles and the time is \(\frac{2}{3}\) hours. Therefore, \( average \, speed = \frac{distance}{time} = \frac{2 \, miles }{\frac{2}{3} \, hours} = 3 \, miles/hour \).

Key Concepts

Converting Minutes to Hours

Understanding how to convert minutes to hours is essential in the context of calculating average speed. Speed is typically expressed in miles per hour, so if we have a time in minutes, it cannot be plugged directly into the speed formula. For this conversion, remember there are 60 minutes in one hour. To convert minutes to hours, you divide the number of minutes by 60.

For example, in the exercise where a swimmer travels 2 miles in 40 minutes, the conversion to hours is done by dividing 40 by 60, which simplifies to \(\frac{2}{3}\) hours or approximately 0.67 hours. This step is crucial; skipping it would lead to incorrect speed calculations. Moreover, understanding this concept helps in daily life—like converting cooking time from recipes or calculating parking rates.

Distance Over Time

The concept of 'distance over time' plays a central role in understanding motion and transportation. It indicates the amount of distance traveled in a given period of time and is the fundamental idea behind the concept of speed.

As we often witness, road signs, vehicle speedometers, and travel itineraries present speed in terms of distance over time, such as miles per hour (mph) or kilometers per hour (kph). A clear grasp of this concept allows us to comprehend not just how to calculate average speed, but also to estimate travel times and distances. In practical scenarios, such as the given exercise where the swimmer covers 2 miles in \(\frac{2}{3}\) hours, we observe 'distance over time' translated into the calculation for average speed.

Speed Calculations

Speed calculations involve dividing the total distance traveled by the total time taken to cover that distance. The formula for average speed is remarkably straightforward: \(speed = \frac{distance}{time}\). When working with average speed, it's important to ensure that distance and time are in consistent units—miles with hours, or kilometers with hours, for example.

In our exercise, after converting 40 minutes to \(\frac{2}{3}\) hours, we calculate the swimmer's average speed by dividing the distance of 2 miles by \(\frac{2}{3}\) hours to get 3 mph. This average speed tells us how fast the swimmer was going on average during their swim. Remember that average speed does not take into account variations in speed during the swim—it is simply a total distance over total time calculation.