Finding The Gym Location A Math Problem Solved

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Let's embark on a fascinating mathematical journey to pinpoint the location of a gym situated between Ping's and Ari's homes. This exploration will not only test our understanding of distance and fractions but also showcase how mathematics can be applied to solve real-world problems. We'll delve into the details of their locations, calculate the distance between their homes, and then determine the precise spot where the gym is located. Let's begin this intriguing quest!

Understanding the Problem

The problem presents us with a scenario where two individuals, Ping and Ari, live at different locations, and a gym is situated somewhere between their homes. Ping resides at the corner of 3rd Street and 6th Avenue, while Ari's abode is at the intersection of 21st Street and 18th Avenue. The crucial piece of information is that the gym is located 23\frac{2}{3} of the distance from Ping's home to Ari's. Our objective is to determine the exact location of the gym.

To solve this problem effectively, we need to make a few assumptions to simplify the calculations. Let's assume that the streets and avenues form a perfect grid, with each block representing a unit distance. This assumption allows us to use the concept of coordinate geometry to represent the locations of Ping's and Ari's homes as points on a coordinate plane. Ping's home can be represented as (3, 6), and Ari's home as (21, 18). The gym's location will then be a point that lies 23\frac{2}{3} of the way along the line segment connecting these two points. This simplification allows us to use mathematical formulas to accurately calculate the gym's location. The assumptions make the problem more tractable and provide a clear path to the solution.

Calculating the Distance

Before we can find the gym's location, we need to determine the distance between Ping's and Ari's homes. In coordinate geometry, the distance between two points (x1, y1) and (x2, y2) can be calculated using the distance formula:

Distance = (x2−x1)2+(y2−y1)2\sqrt{(x2 - x1)^2 + (y2 - y1)^2}.

In our case, Ping's home is at (3, 6), and Ari's home is at (21, 18). Plugging these values into the distance formula, we get:

Distance = (21−3)2+(18−6)2\sqrt{(21 - 3)^2 + (18 - 6)^2} = (18)2+(12)2\sqrt{(18)^2 + (12)^2} = 324+144\sqrt{324 + 144} = 468\sqrt{468}.

Simplifying the square root, we find that 468\sqrt{468} = 36∗13\sqrt{36 * 13} = 613\sqrt{13}. Therefore, the distance between Ping's and Ari's homes is 613\sqrt{13} units. This calculation gives us a precise measure of how far apart the two residences are, which is crucial for determining the gym's location. The distance formula provides a straightforward method for calculating the straight-line distance between two points in a coordinate plane, and it is a fundamental concept in geometry and many applications.

Finding the Gym's Location

Now that we know the distance between Ping's and Ari's homes, we can determine the gym's location. The gym is located 23\frac{2}{3} of the distance from Ping's home to Ari's. To find this point, we can use the section formula, which is a method for finding a point that divides a line segment in a given ratio. If we have two points, P(x1, y1) and Q(x2, y2), and we want to find a point R(x, y) that divides the line segment PQ in the ratio m:n, the coordinates of R are given by:

x = m∗x2+n∗x1m+n\frac{m * x2 + n * x1}{m + n} y = m∗y2+n∗y1m+n\frac{m * y2 + n * y1}{m + n}

In our problem, Ping's home P is (3, 6), Ari's home Q is (21, 18), and the ratio m:n is 2:1 (since the gym is 23\frac{2}{3} of the way from Ping's to Ari's, it is 23\frac{2}{3} of the distance from Ping and 13\frac{1}{3} from Ari). Plugging these values into the section formula, we get:

x = 2∗21+1∗32+1\frac{2 * 21 + 1 * 3}{2 + 1} = 42+33\frac{42 + 3}{3} = 453\frac{45}{3} = 15 y = 2∗18+1∗62+1\frac{2 * 18 + 1 * 6}{2 + 1} = 36+63\frac{36 + 6}{3} = 423\frac{42}{3} = 14

Thus, the gym's location is at the point (15, 14). This means the gym is situated at the corner of 15th Street and 14th Avenue. The section formula provides a precise and elegant way to find a point that divides a line segment in a specific ratio, making it an invaluable tool in coordinate geometry and various practical applications.

Conclusion

In conclusion, by applying mathematical principles such as the distance formula and the section formula, we have successfully determined the location of the gym. The gym is situated at the corner of 15th Street and 14th Avenue. This exercise demonstrates the power of mathematics in solving real-world problems. From understanding the problem and making necessary assumptions to applying the correct formulas and interpreting the results, each step showcases the importance of mathematical reasoning. The journey from Ping's home to Ari's, with the gym marking a significant point along the way, illustrates how mathematical concepts can be used to navigate and understand our spatial environment. This problem not only provides a practical application of mathematical formulas but also enhances our problem-solving skills and logical thinking.

SEO Keywords

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  • Distance Formula: This is a core concept used in the calculation.
  • Section Formula: Another key formula used to find the gym's location.
  • Coordinate Geometry: The branch of mathematics used to solve the problem.
  • Real-World Math Problems: Highlights the practical application of the concepts.
  • Finding Distance: A key aspect of the problem-solving process.
  • Gym Location: The specific goal of the problem.
  • Ping and Ari: The individuals in the scenario, useful for specific searches.
  • Mathematical Problem Solving: The overall skill being demonstrated.
  • Street and Avenue: Contextual keywords related to the location.

Article Title Optimization

  • Original Title: Ping lives at the corner of 3rd Street and 6th Avenue. Ari lives at the corner of 21st Street and 18th Avenue. There is a gym 23\frac{2}{3} the distance from Ping's home to Ari's
  • SEO Title: Finding the Gym Location A Math Problem Solved

Keywords Improvement

  • Original Keywords: mathematics
  • Improved Keywords: Coordinate Geometry, Distance Formula, Section Formula, Real-World Math Problem, Gym Location Problem