A Snowboarder Starts At The Top Of A Mountain That Is 2,500 Feet Above Sea Level. The Snowboarder Then Descends At A Constant Rate Of 650 Feet Per Minute.Which Equation Represents This Situation, Where Y Y Y Represents The Current Altitude Of

Introduction

In this article, we will explore the relationship between altitude and time for a snowboarder descending a mountain. We will use mathematical equations to model this situation and determine the equation that represents the snowboarder's altitude over time.

The Problem

A snowboarder starts at the top of a mountain that is 2,500 feet above sea level. The snowboarder then descends at a constant rate of 650 feet per minute. We want to find the equation that represents the snowboarder's current altitude, denoted by $y$, as a function of time, denoted by $t$.

Modeling the Situation

To model this situation, we can use the concept of linear motion. Since the snowboarder is descending at a constant rate, we can assume that the altitude decreases linearly with time. Let's denote the initial altitude as $y_0$ and the rate of descent as $r$. We can then write the equation for the snowboarder's altitude as a function of time:

$$y(t) = y_0 - rt$$

where $y(t)$ is the current altitude at time $t$, $y_0$ is the initial altitude, and $r$ is the rate of descent.

Substituting the Given Values

We are given that the initial altitude is 2,500 feet and the rate of descent is 650 feet per minute. We can substitute these values into the equation:

$$y(t) = 2500 - 650t$$

Interpreting the Equation

The equation $y(t) = 2500 - 650t$ represents the snowboarder's current altitude as a function of time. At time $t = 0$, the snowboarder is at the top of the mountain, and the altitude is 2,500 feet. As time increases, the altitude decreases linearly, with a rate of 650 feet per minute.

Graphing the Equation

We can graph the equation $y(t) = 2500 - 650t$ to visualize the relationship between altitude and time. The graph will be a straight line with a negative slope, indicating that the altitude decreases linearly with time.

Conclusion

In this article, we used mathematical equations to model the situation of a snowboarder descending a mountain. We found that the equation $y(t) = 2500 - 650t$ represents the snowboarder's current altitude as a function of time. This equation can be used to determine the snowboarder's altitude at any given time, and it provides a mathematical model for the relationship between altitude and time.

Additional Examples

1. Snowboarder Descending at a Different Rate: Suppose the snowboarder descends at a rate of 800 feet per minute. How would the equation change?
2. Snowboarder Climbing a Mountain: Suppose the snowboarder is climbing a mountain at a rate of 200 feet per minute. How would the equation change?
3. Snowboarder Descending a Mountain with a Change in Rate: Suppose the snowboarder descends a mountain at a rate of 650 feet per minute for the first 10 minutes, and then the rate changes to 400 feet per minute. How would the equation change?

Solutions to Additional Examples

1. Snowboarder Descending at a Different Rate: The equation would change to $y(t) = 2500 - 800t$.
2. Snowboarder Climbing a Mountain: The equation would change to $y(t) = 2500 + 200t$.
3. Snowboarder Descending a Mountain with a Change in Rate: The equation would change to $y(t) = 2500 - 650t$ for $0 \leq t \leq 10$, and $y(t) = 2500 - 650(10) + 400(t-10)$ for $t > 10$.

Conclusion

Introduction

In our previous article, we explored the relationship between altitude and time for a snowboarder descending a mountain. We used mathematical equations to model this situation and determined the equation that represents the snowboarder's altitude over time. In this article, we will answer some frequently asked questions about this topic.

Q: What is the initial altitude of the snowboarder?

A: The initial altitude of the snowboarder is 2,500 feet above sea level.

Q: What is the rate of descent of the snowboarder?

A: The rate of descent of the snowboarder is 650 feet per minute.

Q: How can I determine the snowboarder's altitude at a given time?

A: To determine the snowboarder's altitude at a given time, you can use the equation $y(t) = 2500 - 650t$, where $y(t)$ is the current altitude at time $t$.

Q: What happens if the snowboarder descends at a different rate?

A: If the snowboarder descends at a different rate, the equation would change to $y(t) = 2500 - rt$, where $r$ is the new rate of descent.

Q: Can I use this equation to model a snowboarder climbing a mountain?

A: Yes, you can use this equation to model a snowboarder climbing a mountain by changing the sign of the rate of ascent. For example, if the snowboarder climbs at a rate of 200 feet per minute, the equation would be $y(t) = 2500 + 200t$.

Q: What if the snowboarder's rate of descent changes over time?

A: If the snowboarder's rate of descent changes over time, you would need to use a piecewise function to model the situation. For example, if the snowboarder descends at a rate of 650 feet per minute for the first 10 minutes, and then the rate changes to 400 feet per minute, the equation would be $y(t) = 2500 - 650t$ for $0 \leq t \leq 10$, and $y(t) = 2500 - 650(10) + 400(t-10)$ for $t > 10$.

Q: Can I use this equation to model a snowboarder's altitude in a different location?

A: Yes, you can use this equation to model a snowboarder's altitude in a different location by changing the initial altitude and the rate of descent. For example, if the snowboarder is at the top of a mountain that is 3,000 feet above sea level, and descends at a rate of 700 feet per minute, the equation would be $y(t) = 3000 - 700t$.

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

A: Some real-world applications of this equation include:

• Modeling the altitude of a plane or a helicopter over time
• Determining the time it takes for a person to climb a mountain
• Calculating the distance traveled by a person or an object over time
• Modeling the motion of a projectile or a vehicle

Conclusion

In this article, we answered some frequently asked questions about the relationship between altitude and time for a snowboarder descending a mountain. We also provided examples and explanations to help illustrate the application of this concept.