Understanding Average Speed and Calculating Distance in Real-life Scenarios

When it comes to driving to and from work, understanding the relationship between speed and time can be crucial. A common task that many of us face is calculating the distance to our destination based on average speed and total travel time. We'll explore a real-life problem and walk through the steps to solve it, ensuring we optimize our content for google search algorithms.

The Problem: James' Commute

James drives to his office at 32 mph on his return trip. His average speed for the entire trip is 24 mph. If his total trip time is 7 hours, how far is it to his office?

Formula for Average Speed

The formula for average speed is given by the equation:

Avg speed 2ab / (a b)

Where a and b are the speeds of the two segments of the trip. In this case, we have the average speed (24 mph) and one of the segment speeds (32 mph). Our goal is to find the distance to his office using these variables.

Step-by-Step Solution

1. Given: Avg speed 24 mph, speed of return trip (b) 32 mph, total trip time (Tt) 7 hours, and a segment speed (a) 32 mph.

2. Substitute these values into the formula for average speed:

24 2 * 32 * b / (32 b)

3. Simplify and solve for b:

24 64b / (32 b)

4. Cross-multiply to solve for b:

24 * (32 b) 64b

768 24b 64b

768 64b - 24b

768 40b

b 768 / 40

b 19.2

5. Since b is the speed of the return trip, the distance to his office can be calculated using the formula distance speed * time. The time taken for the return trip is 19.2 miles / 32 mph.

6. Time for return trip t1 19.2 / 32 0.6 hours

7. Total trip time 7 hours t1 t2, where t2 is the time for the trip to the office.

8. 7 0.6 t2

9. t2 7 - 0.6

10. t2 6.4 hours

11. Distance to office 32 mph * 6.4 hours 204.8 miles / 14 miles (approximation based on the total distance)

Conclusion

James' commute involves a total distance of approximately 14 miles. The calculation utilizes the concept of average speed and the relationship between speed, time, and distance. By breaking down the problem and understanding the formula, we can ensure accurate calculations and efficiently manage our travel time.

SEO Optimization Tips

To optimize this content for Google search, we've included:

Keywords in the title and within the text, such as "average speed," "distance calculation," and "math problem." Header tags (H1, H2, H3) for structure and readability. Use of specific numbers and calculations to provide concrete examples. Step-by-step instructions for clarity.

This approach ensures our content is search-friendly and provides valuable information to readers searching for similar problems and solutions.