12x L11 P11 F13: Intermediate Algebra - Applications of Rational Functions

Summary

This video explores the application of rational functions using a practical scenario of a road trip from Phoenix to San Diego. It demonstrates how to derive the time of travel as a function of rate, T(r) = 354.5/r. The narrator provides insights on graphing the function, evaluating travel times at different speeds, and analyzing the relationship between rate and time. Key takeaways include how increasing speed reduces travel time, while approaching zero speed leads to increased travel time. The content effectively illustrates the mathematical principles behind distance, rate, and time.

Takeaways

• 😀 The distance from Phoenix to San Diego is 354.5 miles, and the relationship between distance, rate, and time is expressed as d = rt.
• 😀 A rational function T(r) is defined as T = d/r, which represents time as a function of rate.
• 😀 The function can be rewritten as T(r) = 354.5/r, where 354.5 is a constant distance.
• 😀 When graphing T(r), it's essential to restrict values to the first quadrant, focusing on positive rates and times.
• 😀 Recommended graphing window settings include x-values from 0 to 80 and y-values from 0 to 500.
• 😀 If traveling at an average speed of 60 miles per hour, the trip takes approximately 5.9 hours.
• 😀 If the trip takes 10 hours, the average rate of travel is calculated as 35.45 miles per hour.
• 😀 As the rate of travel (r) increases, the time (T) decreases, indicating a direct inverse relationship.
• 😀 Conversely, as the rate approaches zero, the time taken increases, demonstrating that slower speeds require more time.
• 😀 The relationship between rate and time can be explored through various scenarios using the rational function.

Q & A

What is the relationship defined in the transcript for distance, rate, and time?

The relationship is defined by the formula distance equals rate times time (d = rt).

How is the rational function T of r constructed?

T of r is constructed by rearranging the formula to T = d/r, where d is the constant distance of 354.5 miles.

What does the variable 'r' represent in the function T of r?

'r' represents the rate of travel in miles per hour.

What are the recommended graphing window values for the function T of r?

The recommended window values are: Xmin = 0, Xmax = 80, Ymin = 0, and Ymax = 500.

If traveling at 60 miles per hour, how long will the trip take?

Traveling at 60 miles per hour will take approximately 5.9 hours for the trip.

How can the average rate of travel be determined if the trip took 10 hours?

The average rate can be calculated by rearranging the function to r = 354.5/10, which gives r approximately 35.45 miles per hour.

What does the graph indicate about the relationship between rate and time?

The graph indicates that as the rate increases, the time decreases, meaning faster travel results in shorter trip durations.

What happens to time as the rate approaches zero?

As the rate approaches zero, the time increases, indicating that slower travel takes longer.

Why is it important to stay in the first quadrant when graphing T of r?

It's important to stay in the first quadrant because negative values for time or rate do not make sense in this context.

What is the significance of using a constant distance of 354.5 miles in the function?

The constant distance of 354.5 miles serves as a fixed reference point to analyze how variations in rate affect travel time.