Joseph And Isabelle Left Omyra's House At The Same Time. Joseph Jogged North At 8 Kilometers Per Hour, While Isabelle Rode Her Bike West At 12 Kilometers Per Hour. Omyra Tried To Figure Out How Far Apart They Were After 1.5 Hours. Her Work Is Shown
Introduction
In the world of mathematics, problems often arise from everyday situations. One such scenario involves Joseph and Isabelle, who left Omyra's house at the same time and traveled in different directions. Joseph jogged north at a speed of 8 kilometers per hour, while Isabelle rode her bike west at a speed of 12 kilometers per hour. The question on Omyra's mind was: how far apart were they after 1.5 hours? In this article, we will delve into the mathematical concepts that govern this problem and explore the solution.
The Problem
Joseph and Isabelle left Omyra's house at the same time. Joseph jogged north at a speed of 8 kilometers per hour, while Isabelle rode her bike west at a speed of 12 kilometers per hour. We need to find the distance between them after 1.5 hours.
The Solution
To solve this problem, we can use the concept of relative motion. Since Joseph and Isabelle are moving in different directions, we need to consider their relative speeds. We can use the Pythagorean theorem to find the distance between them.
Let's break down the problem step by step:
- Find the distance traveled by Joseph: Since Joseph jogged north at a speed of 8 kilometers per hour, we can find the distance traveled by him after 1.5 hours using the formula:
Distance = Speed × Time = 8 km/h × 1.5 h = 12 km
- Find the distance traveled by Isabelle: Since Isabelle rode her bike west at a speed of 12 kilometers per hour, we can find the distance traveled by her after 1.5 hours using the formula:
Distance = Speed × Time = 12 km/h × 1.5 h = 18 km
- Use the Pythagorean theorem to find the distance between them: Since Joseph and Isabelle are moving in different directions, we can use the Pythagorean theorem to find the distance between them. The Pythagorean theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the distance between Joseph and Isabelle is the hypotenuse, and the distances traveled by them are the other two sides. We can write the Pythagorean theorem as:
c² = a² + b²
where c is the distance between Joseph and Isabelle, a is the distance traveled by Joseph, and b is the distance traveled by Isabelle.
Plugging in the values, we get:
c² = 12² + 18² = 144 + 324 = 468
Taking the square root of both sides, we get:
c = √468 = 21.6 km
Therefore, the distance between Joseph and Isabelle after 1.5 hours is approximately 21.6 kilometers.
Conclusion
In this article, we explored the mathematical concepts that govern the problem of Joseph and Isabelle traveling in different directions. We used the concept of relative motion and the Pythagorean theorem to find the distance between them after 1.5 hours. The solution involved breaking down the problem into smaller steps and using mathematical formulas to find the distance traveled by each person. The final answer was approximately 21.6 kilometers.
Real-World Applications
The concept of relative motion and the Pythagorean theorem have numerous real-world applications. For example:
- Navigation: Understanding relative motion is crucial for navigation, especially in situations where multiple objects are moving in different directions.
- Physics: The Pythagorean theorem is used to describe the motion of objects in physics, including the motion of projectiles and the motion of objects in circular motion.
- Engineering: The Pythagorean theorem is used in engineering to design and build structures, such as bridges and buildings.
Final Thoughts
Introduction
In our previous article, we explored the mathematical concepts that govern the problem of Joseph and Isabelle traveling in different directions. We used the concept of relative motion and the Pythagorean theorem to find the distance between them after 1.5 hours. In this article, we will answer some of the most frequently asked questions about this problem.
Q&A
Q: What is the concept of relative motion?
A: Relative motion is the concept of motion in relation to a reference frame. In the context of Joseph and Isabelle, relative motion refers to the motion of each person in relation to the other.
Q: Why is the Pythagorean theorem used to find the distance between Joseph and Isabelle?
A: The Pythagorean theorem is used to find the distance between Joseph and Isabelle because it describes the relationship between the lengths of the sides of a right-angled triangle. In this case, the distance between Joseph and Isabelle is the hypotenuse of the triangle, and the distances traveled by each person are the other two sides.
Q: What is the significance of the Pythagorean theorem in real-world applications?
A: The Pythagorean theorem has numerous real-world applications, including navigation, physics, and engineering. It is used to describe the motion of objects, design and build structures, and solve problems involving right-angled triangles.
Q: How can I apply the concept of relative motion to real-world situations?
A: The concept of relative motion can be applied to real-world situations by considering the motion of objects in relation to a reference frame. For example, if you are driving a car and another car is passing you, you can use the concept of relative motion to determine the speed and direction of the other car.
Q: What is the difference between relative motion and absolute motion?
A: Relative motion refers to the motion of an object in relation to a reference frame, while absolute motion refers to the motion of an object in relation to a fixed point. For example, if you are standing on a train and it is moving at a speed of 60 km/h, your relative motion is 60 km/h, but your absolute motion is 0 km/h if you are standing still.
Q: Can I use the Pythagorean theorem to find the distance between two objects that are moving in different directions?
A: Yes, you can use the Pythagorean theorem to find the distance between two objects that are moving in different directions. However, you need to consider the relative motion of the objects and use the concept of relative motion to determine the distance between them.
Q: What is the formula for finding the distance between two objects that are moving in different directions?
A: The formula for finding the distance between two objects that are moving in different directions is:
c² = a² + b²
where c is the distance between the objects, a is the distance traveled by one object, and b is the distance traveled by the other object.
Q: Can I use the Pythagorean theorem to find the distance between two objects that are moving in the same direction?
A: Yes, you can use the Pythagorean theorem to find the distance between two objects that are moving in the same direction. However, you need to consider the relative motion of the objects and use the concept of relative motion to determine the distance between them.
Q: What is the formula for finding the distance between two objects that are moving in the same direction?
A: The formula for finding the distance between two objects that are moving in the same direction is:
c = a + b
where c is the distance between the objects, a is the distance traveled by one object, and b is the distance traveled by the other object.
Conclusion
In this article, we answered some of the most frequently asked questions about the problem of Joseph and Isabelle traveling in different directions. We hope that this article has provided a clear and concise explanation of the mathematical concepts involved and has inspired readers to explore the world of mathematics.