Josh Is Hiking In Glacier National Park. He Has Now Hiked A Total Of 17 Km And Is 2 Km Short Of Being 1 2 \frac{1}{2} 2 1 ​ Of The Way Done With His Hike.1. Write An Equation To Determine The Total Length In Kilometers ( H H H ) Of Josh's Hike.

Introduction

Josh is an avid hiker who has embarked on an exciting adventure in Glacier National Park. As he navigates through the breathtaking landscapes, he has covered a significant distance of 17 km. However, he still has a way to go before completing his hike. In this article, we will delve into the mathematical world to determine the total length of Josh's hike.

The Problem

Josh has hiked a total of 17 km and is 2 km short of being $\frac{1}{2}$ of the way done with his hike. This means that if he covers the remaining 2 km, he will have completed half of his hike. Let's denote the total length of Josh's hike as $h$ km.

Formulating the Equation

To determine the total length of Josh's hike, we need to set up an equation based on the given information. Since Josh is 2 km short of being $\frac{1}{2}$ of the way done with his hike, we can express this as an equation:

$$\frac{1}{2}h - 2 = 17$$

This equation states that half of the total length of the hike ($\frac{1}{2}h$) minus 2 km is equal to 17 km, which is the distance Josh has already covered.

Solving the Equation

To solve for $h$, we need to isolate the variable on one side of the equation. We can start by adding 2 to both sides of the equation:

$$\frac{1}{2}h - 2 + 2 = 17 + 2$$

This simplifies to:

$$\frac{1}{2}h = 19$$

Next, we can multiply both sides of the equation by 2 to eliminate the fraction:

$$2 \times \frac{1}{2}h = 2 \times 19$$

This simplifies to:

$$h = 38$$

Conclusion

Therefore, the total length of Josh's hike is 38 km. This means that Josh has covered 17 km and still has 21 km to go before completing his hike.

Discussion

This problem requires a basic understanding of algebraic equations and fractions. The equation $\frac{1}{2}h - 2 = 17$ represents a linear equation in one variable, where $h$ is the unknown variable. By solving for $h$, we can determine the total length of Josh's hike.

Real-World Applications

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

• Hiking and Outdoor Activities: Understanding the distance and duration of a hike is crucial for planning and executing a successful outdoor adventure.
• Travel and Transportation: Calculating the total distance of a trip is essential for planning routes, estimating travel time, and determining fuel consumption.
• Science and Engineering: Solving equations and calculating distances are fundamental skills in various scientific and engineering disciplines, such as physics, engineering, and computer science.

Tips and Variations

• Variation 1: Suppose Josh has hiked a total of 17 km and is 2 km short of being $\frac{1}{3}$ of the way done with his hike. How would you modify the equation to solve for $h$?
• Variation 2: Suppose Josh has hiked a total of 17 km and is 2 km short of being $\frac{2}{3}$ of the way done with his hike. How would you modify the equation to solve for $h$?

Conclusion

Introduction

In our previous article, we explored the mathematical world of Josh's hike in Glacier National Park. We determined that the total length of Josh's hike is 38 km. In this article, we will delve into a Q&A session to further clarify the concepts and provide additional insights.

Q&A Session

Q: What is the significance of the equation $\frac{1}{2}h - 2 = 17$? A: The equation $\frac{1}{2}h - 2 = 17$ represents a linear equation in one variable, where $h$ is the unknown variable. This equation states that half of the total length of the hike ($\frac{1}{2}h$) minus 2 km is equal to 17 km, which is the distance Josh has already covered.

Q: How did you solve for $h$ in the equation $\frac{1}{2}h - 2 = 17$? A: To solve for $h$, we added 2 to both sides of the equation, which resulted in $\frac{1}{2}h = 19$. Then, we multiplied both sides of the equation by 2 to eliminate the fraction, which resulted in $h = 38$.

Q: What is the total length of Josh's hike? A: The total length of Josh's hike is 38 km.

Q: How much distance is Josh still short of completing his hike? A: Josh is still 2 km short of completing his hike.

Q: What is the significance of the fraction $\frac{1}{2}$ in the equation? A: The fraction $\frac{1}{2}$ represents half of the total length of the hike. This means that if Josh covers the remaining 2 km, he will have completed half of his hike.

Q: Can you provide an example of a real-world application of this problem? A: Yes, understanding the distance and duration of a hike is crucial for planning and executing a successful outdoor adventure. For instance, if Josh is planning to hike a total of 38 km, he needs to ensure that he has enough food, water, and energy to complete the hike.

Q: How would you modify the equation to solve for $h$ if Josh has hiked a total of 17 km and is 2 km short of being $\frac{1}{3}$ of the way done with his hike? A: To modify the equation, we would need to replace the fraction $\frac{1}{2}$ with $\frac{1}{3}$. The new equation would be:

$$\frac{1}{3}h - 2 = 17$$

We would then solve for $h$ by adding 2 to both sides of the equation, which would result in $\frac{1}{3}h = 19$. Then, we would multiply both sides of the equation by 3 to eliminate the fraction, which would result in $h = 57$.

Q: How would you modify the equation to solve for $h$ if Josh has hiked a total of 17 km and is 2 km short of being $\frac{2}{3}$ of the way done with his hike? A: To modify the equation, we would need to replace the fraction $\frac{1}{2}$ with $\frac{2}{3}$. The new equation would be:

$$\frac{2}{3}h - 2 = 17$$

We would then solve for $h$ by adding 2 to both sides of the equation, which would result in $\frac{2}{3}h = 19$. Then, we would multiply both sides of the equation by $\frac{3}{2}$ to eliminate the fraction, which would result in $h = 28.5$.

Conclusion

In conclusion, this Q&A session has provided additional insights into the mathematical world of Josh's hike in Glacier National Park. We have clarified the concepts and provided examples of real-world applications. By understanding the distance and duration of a hike, we can plan and execute a successful outdoor adventure.