Teresa Drove To A Resort 372 Miles From Her Home. She Averaged 51 Mph For The First Part Of Her Trip And 56 Mph For The Second Part. If Her Total Driving Time Was 7 Hours, How Long Did She Travel At Each Rate?Answer:First Part: ____ Hours Second Part:

Mar 12, 2025 by ADMIN 253 views

**Teresa's Road Trip: A Mathematical Puzzle**

Introduction

Teresa embarked on a road trip to a resort, covering a total distance of 372 miles from her home. To solve this problem, we need to break down her journey into two parts, each with a different average speed. The first part of her trip was covered at an average speed of 51 mph, while the second part was at an average speed of 56 mph. Given that her total driving time was 7 hours, we aim to determine how long she traveled at each rate.

Breaking Down the Problem

To solve this problem, we can use the concept of distance, speed, and time. The formula to calculate distance is:

Distance = Speed × Time

We know the total distance (372 miles) and the total time (7 hours). We also know the average speeds for the two parts of the trip (51 mph and 56 mph). Let's denote the time spent traveling at 51 mph as x hours and the time spent traveling at 56 mph as (7 - x) hours.

Calculating Time Spent at Each Speed

We can set up two equations using the distance formula:

For the first part of the trip (51 mph): 51x = Distance (first part)
For the second part of the trip (56 mph): 56(7 - x) = Distance (second part)

Since the total distance is 372 miles, we can set up an equation that combines the two parts:

51x + 56(7 - x) = 372

Solving the Equation

Now, let's solve the equation to find the value of x, which represents the time spent traveling at 51 mph.

51x + 392 - 56x = 372

Combine like terms:

-5x = -20

Divide both sides by -5:

x = 4

Interpreting the Results

Now that we have found the value of x, we can determine the time spent traveling at each speed.

Time spent traveling at 51 mph: 4 hours
Time spent traveling at 56 mph: 7 - 4 = 3 hours

Therefore, Teresa traveled at 51 mph for 4 hours and at 56 mph for 3 hours.

Conclusion

In this problem, we used the concept of distance, speed, and time to solve a mathematical puzzle. By breaking down the problem into two parts and setting up equations, we were able to find the time spent traveling at each speed. This problem demonstrates the importance of using mathematical concepts to solve real-world problems.

Discussion

This problem can be used to discuss various mathematical concepts, such as:

Distance, speed, and time
Algebraic equations
Problem-solving strategies
Critical thinking

Real-World Applications

This problem has real-world applications in various fields, such as:

Transportation: Understanding the relationship between distance, speed, and time is crucial for planning routes, estimating travel times, and optimizing transportation systems.
Logistics: Companies need to calculate distances, speeds, and times to optimize their supply chains and delivery routes.
Engineering: Engineers use mathematical models to design and optimize systems, including transportation systems.

Additional Resources

For further practice and exploration, consider the following resources:

Khan Academy: Distance, Speed, and Time
Mathway: Algebraic Equations
Wolfram Alpha: Mathematical Calculations

Introduction

In our previous article, we solved a mathematical puzzle involving Teresa's road trip to a resort. We used the concept of distance, speed, and time to determine how long she traveled at each rate. In this article, we'll provide a Q&A section to further clarify the concepts and provide additional insights.

Q&A

Q: What is the formula to calculate distance?

A: The formula to calculate distance is:

Distance = Speed × Time

Q: How do we use the distance formula to solve this problem?

A: We use the distance formula to set up two equations, one for each part of the trip. We know the total distance (372 miles) and the total time (7 hours). We also know the average speeds for the two parts of the trip (51 mph and 56 mph).

Q: What is the significance of the time spent traveling at each speed?

A: The time spent traveling at each speed is crucial in determining the distance covered during each part of the trip. By knowing the time spent at each speed, we can calculate the distance covered during each part of the trip.

Q: How do we calculate the time spent traveling at each speed?

A: We can calculate the time spent traveling at each speed by setting up an equation that combines the two parts of the trip. We use the distance formula to set up the equation and then solve for the unknown variable (time).

Q: What is the value of x in the equation?

A: The value of x represents the time spent traveling at 51 mph. We found that x = 4 hours.

Q: What is the time spent traveling at 56 mph?

A: Since the total time is 7 hours, and the time spent traveling at 51 mph is 4 hours, the time spent traveling at 56 mph is 7 - 4 = 3 hours.

Q: What are some real-world applications of this problem?

A: This problem has real-world applications in various fields, such as:

Transportation: Understanding the relationship between distance, speed, and time is crucial for planning routes, estimating travel times, and optimizing transportation systems.
Logistics: Companies need to calculate distances, speeds, and times to optimize their supply chains and delivery routes.
Engineering: Engineers use mathematical models to design and optimize systems, including transportation systems.

Q: What are some additional resources for further practice and exploration?

A: Some additional resources for further practice and exploration include:

Khan Academy: Distance, Speed, and Time
Mathway: Algebraic Equations
Wolfram Alpha: Mathematical Calculations