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Definition: Distance is a numerical measurement of how far apart two points or objects are. It is often represented by the symbol “d” and is measured in units such as meters, kilometers, or miles. In mathematics, the formal definition of distance between two points in Euclidean space is the length of the shortest path connecting the two points. This shortest path is known as a straight line or a line segment.

Examples:

• Find the distance between the points (1, 2) and (4, 6) on a Cartesian plane. To find the distance, we use the distance formula, which is d = sqrt((x2-x1)^2 + (y2-y1)^2). Plugging in the coordinates, we get d = sqrt((4-1)^2 + (6-2)^2) = sqrt(9 + 16) = 5.
• A person walks 10 meters north, then turns and walks 15 meters east. What is the distance the person has traveled? To find the distance traveled, we use the Pythagorean theorem, which states that the sum of the squares of the lengths of the two legs of a right triangle is equal to the square of the length of the hypotenuse. In this case, the distance traveled is the hypotenuse of a right triangle with legs of 10 meters and 15 meters. So, the distance traveled is d = sqrt(10^2 + 15^2) = sqrt(325) = 18.027 meters.
• A car drives 60 miles per hour for 2 hours. What is the distance the car has traveled? To find the distance traveled, we use the formula d = rt, where r is the rate or speed of the car, and t is the time the car has traveled. Plugging in the values, we get d = 60 miles/hour * 2 hours = 120 miles.
• A marathon runner completes a race in 3 hours and 45 minutes, covering a distance of 26.2 miles. What was the runner’s average speed in miles per hour? To find the average speed, we use the formula s = d/t, where s is the speed, d is the distance, and t is the time. Converting the time to hours, we get t = 3.75 hours. Plugging in the values, we get s = 26.2 miles / 3.75 hours = 6.99 miles per hour.
• Two cities are 500 miles apart. If a car drives from one city to the other at a speed of 50 miles per hour, how long does it take to complete the journey? To find the time, we use the formula t = d/r, where t is the time, d is the distance, and r is the rate or speed. Plugging in the values, we get t = 500 miles / 50 miles per hour = 10 hours.

Importance: The concept of distance is important in many fields, including physics, engineering, and geography. In physics, the concept of distance is used to describe the separation between objects in space. It is also used to calculate the amount of work done by a force in moving an object a certain distance. In engineering, distance is used to measure the lengths and sizes of various components and structures. In geography, distance is used to measure the separation between two locations and to calculate travel time and distance.

Quiz:

1. What is the formal definition of distance in Euclidean space?
2. How do you find the distance between two points on a Cartesian plane?
3. What is the formula for finding the distance traveled by a car at a constant speed?
4. If a person walks 5 meters north, then turns and walks 10 meters east, what is the distance the person has traveled?
5. What is the average speed of a car that completes a journey of 150 miles in 3 hours?
6. What is the distance between the points (2, 3) and (5, 8) on a Cartesian plane?
7. How do you find the time it takes for a car to complete a journey at a given speed?
8. What is the Pythagorean theorem?
9. How is the concept of distance used in physics?
10. How is the concept of distance used in analysis?

Answers:

1. The formal definition of distance in Euclidean space is the length of the shortest path connecting two points.
2. To find the distance between two points on a Cartesian plane, use the distance formula: d = sqrt((x2-x1)^2 + (y2-y1)^2).
3. The formula for finding the distance traveled by a car at a constant speed is d = rt, where d is the distance, r is the rate or speed, and t is the time.
4. The distance the person has traveled is the hypotenuse of a right triangle with legs of 5 meters and 10 meters. So, the distance is d = sqrt(5^2 + 10^2) = sqrt(125) = 11.18 meters.
5. The average speed of the car is s = d/t, where d is the distance and t is the time. Plugging in the values, we get s = 150 miles / 3 hours = 50 miles per hour.
6. The distance between the points (2, 3) and (5, 8) is d = sqrt((5-2)^2 + (8-3)^2) = sqrt(9 + 25) = 5*sqrt(2).
7. The time it takes for a car to complete a journey at a given speed is found using the formula t = d/r, where t is the time, d is the distance, and r is the rate or speed.
8. The Pythagorean theorem states that the sum of the squares of the lengths of the two legs of a right triangle is equal to the square of the length of the hypotenuse.
9. In physics, the concept of distance is used to describe the separation between objects in space and to calculate the amount of work done by a force in moving an object a certain distance.
10.  In analysis, distance is used to define the concept of continuity, which is essential for understanding functions and calculus.

Conclusion: Distance is a fundamental concept in mathematics and many other fields. It refers to the amount of space between two points or objects and is often represented by the symbol “d”. The formal definition of distance in Euclidean space is the length of the shortest path connecting two points, but there are many other ways to define distance in different contexts.

In this article, we have explored some of the key concepts related to distance in mathematics. We have seen how distance is used in geometry, topology, and analysis, and we have examined some practical applications of distance, such as measuring travel time and distance in geography, and calculating work done by a force in physics.

We have also provided examples and a quiz to help readers test their understanding of the material. By mastering the concept of distance, students can gain a deeper understanding of many mathematical and scientific principles, and be better equipped to solve problems and make sense of the world around them.

In summary, distance is an essential concept in mathematics and many other fields. By understanding the various ways it is defined and applied, we can gain a deeper understanding of the world around us and the principles that govern it.