Problem 2 - College Algebra with Corequisite Support

Question

Jessica is walking home from a friend's house. After 2 minutes she is 1.4 miles from home. Twelve minutes after leaving, she is 0.9 miles from home. What is her rate in miles per hour?

Step-by-Step Solution

Answer

Jessica's walking rate is 3 miles per hour.

Step 1: Understand the Problem

We need to find Jessica's walking rate in miles per hour. We know her distances from home at two different times. We can use these to determine her speed.

Step 2: Determine Time Taken

The timeline provided includes distances after 2 minutes and 12 minutes. Therefore, the time interval we're looking at is from 2 minutes to 12 minutes, which is 12 - 2 = 10 minutes.

Step 3: Determine Distance Covered

Initially, after 2 minutes, Jessica was 1.4 miles from home. Twelve minutes in, she is 0.9 miles from home. Therefore, the distance she covered during the 10-minute interval is 1.4 - 0.9 = 0.5 miles.

Step 4: Calculate Walking Rate in Miles per Minute

Jessica covered 0.5 miles in 10 minutes. We find the rate in miles per minute by dividing the distance by time: 0.5 miles / 10 minutes = 0.05 miles per minute.

Step 5: Convert Rate to Miles per Hour

To find the rate in miles per hour, we convert 0.05 miles per minute to hours. Since there are 60 minutes in an hour, we multiply 0.05 by 60. Thus, 0.05 miles/min * 60 min/hour = 3 miles per hour.

Key Concepts

Time Interval Calculation

When solving a problem involving rates, like Jessica's walking rate, identifying the correct time interval is crucial. In the given problem, we start by marking the crucial times in Jessica's journey. We note that one distance measurement is taken after 2 minutes and another after 12 minutes.

To find the time interval she was walking, we subtract the starting time from the ending time. Here, it looks like this:

• Ending Time: 12 minutes
• Starting Time: 2 minutes
• Time Interval: 12 - 2 = 10 minutes

This simple subtraction gives us a 10-minute time interval during which Jessica covered a certain distance. Understanding how to calculate this interval correctly is foundational, as it impacts the distance measurement and rate calculations that follow.

Distance Measurement

Next, let's examine how distance measurements help us understand the journey. In this problem, Jessica's distance from her home gives us clues about the distance she traveled over the time interval. We know that:

• At 2 minutes, she was 1.4 miles away from her home.
• At 12 minutes, she was 0.9 miles away.

To figure out how far she actually covered in those 10 minutes, we perform a subtraction with these distances: 1.4 miles - 0.9 miles = 0.5 miles.

This means Jessica walked 0.5 miles in that specific interval. This type of distance measurement, guiding us to look at changes instead of absolute values, helps us find out how much ground was covered during a period of movement.

Rate Conversion

The final step involves understanding and calculating the rate of movement in a more conventional unit, like miles per hour. Initially, Jessica's walking speed was found in miles per minute. We calculated that she walked 0.5 miles in 10 minutes, resulting in a rate of: 0.5 miles / 10 minutes = 0.05 miles per minute.

However, we are often more familiar with rates per hour, such as miles per hour (mph), so a conversion is needed. To convert from minutes to hours, we use the fact that there are 60 minutes in an hour:

• Rate in miles per minute: 0.05
• Conversion factor: 60 minutes/hour

By multiplying: 0.05 miles/minute * 60 minutes/hour = 3 miles/hour.

This calculated rate tells us that if Jessica maintained this pace, she would walk 3 miles in an hour. Converting rates this way is a handy technique to express speeds in more comprehensible and commonly used measurements.