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Part 1
A railway system on a hillside moves passengers at a constant rate to an elevation of 50 m. The elevations of a train are given for 2 different locations.
a. Write an equation in point-slope form to represent the elevation of the train in terms of time. How can the equation be used to find the rate of increase in elevation of the train in meters per second?
b. At what elevation does the train start initially? Write a linear equation in a form that gives the information as part of the equation. Explain your reasoning.
Part 1
A railway system on a hillside moves passengers at a constant rate to an elevation of 50 m. The elevations of a train are given for 2 different locations.
a. Write an equation in point-slope form to represent the elevation of the train in terms of time. How can the equation be used to find the rate of increase in elevation of the train in meters per second?
b. At what elevation does the train start initially? Write a linear equation in a form that gives the information as part of the equation. Explain your reasoning.
Answers
GPT-4o mini
To address the questions posed, we'll break this down into the two parts you've presented. Let's first consider some details we might have, although no specific numerical points on the elevation or time are given in your original text.
### Part 1: Point-Slope Form and Rate of Increase
**a. Write an equation in point-slope form.**
In point-slope form, the equation of a line can be expressed as:
\[
y - y_1 = m(x - x_1)
\]
Where:
- \( y \) is the dependent variable (elevation in this case).
- \( x \) is the independent variable (time).
- \( m \) is the slope (which represents the rate of increase in elevation over time).
- \((x_1, y_1)\) is a specific point on the line.
Let's assume from the problem context that we have two specific points regarding time and elevation:
1. Let's say at \( t = 0 \) seconds, \( \text{Elevation} = E_0 \) (perhaps this is the train's starting elevation).
2. At \( t = t_1 \) seconds, \( \text{Elevation} = E_1 = 50 \) m (the peak height reached).
To express this in point-slope form, we start by picking our initial point. If we let \( (0, E_0) \) be the starting point and \( (t_1, 50) \) be the endpoint, the rate of change \( m \) or slope can be calculated as:
\[
m = \frac{50 - E_0}{t_1 - 0} = \frac{50 - E_0}{t_1}
\]
Substituting into the point-slope equation (assuming we know \( E_0 \) and \( t_1 \)) gives:
\[
y - E_0 = \frac{50 - E_0}{t_1} (x - 0)
\]
This can be rewritten as:
\[
y - E_0 = \frac{50 - E_0}{t_1}x
\]
**Finding the rate of increase in elevation:** The slope \( m \), which is \( \frac{50 - E_0}{t_1} \), represents the rate of increase in elevation of the train in meters per second.
### Part 2: Initial Elevation and Linear Equation
**b. Initial Elevation and Linear Equation**
The initial elevation of the train, based on our points, is \( E_0 \) at \( t = 0 \). If no specific value has been provided for the initial elevation, you might need to clarify it based on additional context or established norms for such a railway.
A linear equation that captures this information could be simply derived from the two points we have. If we assume \( E_0 \) is the starting point, the linear equation in slope-intercept form (where we express \( y \) in terms of \( x \)) would look like:
\[
y = mx + b
\]
Where \( b = E_0 \) (the y-intercept when \( x = 0 \)). Therefore, if in point-slope we have established that the rate is included, it shows the relationship clearly:
\[
y = \frac{50 - E_0}{t_1} x + E_0
\]
### Final Response
1. **The point-slope equation** expresses the elevation in terms of time, allowing us to determine the rate of increase in elevation.
2. **The initial elevation** is represented in the linear equation where \( b \) directly reflects this initial point, and the equation clearly outlines the manner in which elevation changes over time until it reaches 50 m.
If specific values for \( E_0 \) and \( t_1 \) are known, more tailored and exact equations can be provided.
### Part 1: Point-Slope Form and Rate of Increase
**a. Write an equation in point-slope form.**
In point-slope form, the equation of a line can be expressed as:
\[
y - y_1 = m(x - x_1)
\]
Where:
- \( y \) is the dependent variable (elevation in this case).
- \( x \) is the independent variable (time).
- \( m \) is the slope (which represents the rate of increase in elevation over time).
- \((x_1, y_1)\) is a specific point on the line.
Let's assume from the problem context that we have two specific points regarding time and elevation:
1. Let's say at \( t = 0 \) seconds, \( \text{Elevation} = E_0 \) (perhaps this is the train's starting elevation).
2. At \( t = t_1 \) seconds, \( \text{Elevation} = E_1 = 50 \) m (the peak height reached).
To express this in point-slope form, we start by picking our initial point. If we let \( (0, E_0) \) be the starting point and \( (t_1, 50) \) be the endpoint, the rate of change \( m \) or slope can be calculated as:
\[
m = \frac{50 - E_0}{t_1 - 0} = \frac{50 - E_0}{t_1}
\]
Substituting into the point-slope equation (assuming we know \( E_0 \) and \( t_1 \)) gives:
\[
y - E_0 = \frac{50 - E_0}{t_1} (x - 0)
\]
This can be rewritten as:
\[
y - E_0 = \frac{50 - E_0}{t_1}x
\]
**Finding the rate of increase in elevation:** The slope \( m \), which is \( \frac{50 - E_0}{t_1} \), represents the rate of increase in elevation of the train in meters per second.
### Part 2: Initial Elevation and Linear Equation
**b. Initial Elevation and Linear Equation**
The initial elevation of the train, based on our points, is \( E_0 \) at \( t = 0 \). If no specific value has been provided for the initial elevation, you might need to clarify it based on additional context or established norms for such a railway.
A linear equation that captures this information could be simply derived from the two points we have. If we assume \( E_0 \) is the starting point, the linear equation in slope-intercept form (where we express \( y \) in terms of \( x \)) would look like:
\[
y = mx + b
\]
Where \( b = E_0 \) (the y-intercept when \( x = 0 \)). Therefore, if in point-slope we have established that the rate is included, it shows the relationship clearly:
\[
y = \frac{50 - E_0}{t_1} x + E_0
\]
### Final Response
1. **The point-slope equation** expresses the elevation in terms of time, allowing us to determine the rate of increase in elevation.
2. **The initial elevation** is represented in the linear equation where \( b \) directly reflects this initial point, and the equation clearly outlines the manner in which elevation changes over time until it reaches 50 m.
If specific values for \( E_0 \) and \( t_1 \) are known, more tailored and exact equations can be provided.