Question
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. If x is the time in seconds and y is the elevation of the train in​ meters, then (enter your response here) represents the elevation of the train in terms of time.
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.
a. If x is the time in seconds and y is the elevation of the train in​ meters, then (enter your response here) represents the elevation of the train in terms of time.
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 regarding the railway system:
### Part a:
Let's denote the elevation of the train at time \(x\) seconds as a function \(y(x)\). Since the train is moving at a constant rate to an elevation of 50 m, we can represent the elevation of the train as:
\[
y(x) = mx + b
\]
Where:
- \(m\) is the rate of elevation change (slope),
- \(b\) is the initial elevation (y-intercept).
However, since we don't have specific numerical values for \(m\) or information about how long it takes to reach the elevation of 50 m, we can't represent the equation numerically.
If we assume the train reaches 50 m at a certain time \(T\) (let's say it takes \(T\) seconds to reach 50 m), then the linear equation could be represented in a general way as:
\[
y(x) = \frac{50}{T} x
\]
This equation shows that the elevation increases linearly as time progresses until reaching 50 m, assuming the train starts at 0 m.
### Part b:
To find the initial elevation of the train, we consider the linear equation \(y(x) = mx + b\).
The initial elevation occurs at \(x = 0\). Thus, to find the elevation when the train starts, we calculate:
\[
y(0) = b
\]
Since we haven't specified any initial elevation in the problem, we need to set an initial condition. If we assume the train starts from ground level, the initial elevation \(b\) would be 0. Assuming 0 m as the starting point, we can present the linear equation as:
\[
y(x) = \frac{50}{T} x + 0
\]
Where \(b = 0\), indicating that the train starts from an elevation of 0 m.
In conclusion, the reasoning for determining the initial elevation in this case is based on the typical scenario where the train starts at ground level. If the train starts at a different elevation (like a station already elevated), we'd need that information to specify \(b\) accordingly.
### Part a:
Let's denote the elevation of the train at time \(x\) seconds as a function \(y(x)\). Since the train is moving at a constant rate to an elevation of 50 m, we can represent the elevation of the train as:
\[
y(x) = mx + b
\]
Where:
- \(m\) is the rate of elevation change (slope),
- \(b\) is the initial elevation (y-intercept).
However, since we don't have specific numerical values for \(m\) or information about how long it takes to reach the elevation of 50 m, we can't represent the equation numerically.
If we assume the train reaches 50 m at a certain time \(T\) (let's say it takes \(T\) seconds to reach 50 m), then the linear equation could be represented in a general way as:
\[
y(x) = \frac{50}{T} x
\]
This equation shows that the elevation increases linearly as time progresses until reaching 50 m, assuming the train starts at 0 m.
### Part b:
To find the initial elevation of the train, we consider the linear equation \(y(x) = mx + b\).
The initial elevation occurs at \(x = 0\). Thus, to find the elevation when the train starts, we calculate:
\[
y(0) = b
\]
Since we haven't specified any initial elevation in the problem, we need to set an initial condition. If we assume the train starts from ground level, the initial elevation \(b\) would be 0. Assuming 0 m as the starting point, we can present the linear equation as:
\[
y(x) = \frac{50}{T} x + 0
\]
Where \(b = 0\), indicating that the train starts from an elevation of 0 m.
In conclusion, the reasoning for determining the initial elevation in this case is based on the typical scenario where the train starts at ground level. If the train starts at a different elevation (like a station already elevated), we'd need that information to specify \(b\) accordingly.