Question
Mt. Eglis has an aerial gondola that carries visitors up and down the mountain. At the top of the mountain, the gondola reaches an elevation of 9,100 feet. On the way down, the gondola descends at a rate of 240 feet per minute. You can use a function to describe the elevation of the gondola x minutes after it starts descending. Is the function linear or exponential? Which equation represents the function?
Answers
GPT-4o mini
The scenario described involves a gondola descending from a specific elevation at a constant rate. Since the gondola descends at a steady rate of 240 feet per minute, the change in elevation over time is linear.
A linear function can be described by the equation:
\[ y = mx + b \]
where:
- \( y \) is the elevation of the gondola,
- \( m \) is the rate of change (slope),
- \( x \) is the time in minutes,
- \( b \) is the initial value (elevation at the start of descent).
In this case:
- The initial elevation \( b \) is 9,100 feet,
- The rate of descent \( m \) is -240 feet per minute (since the elevation is decreasing).
Thus, the equation representing the elevation of the gondola \( y \) after \( x \) minutes of descent can be written as:
\[ y = -240x + 9100 \]
This equation is linear and models the gondola’s descent effectively.
A linear function can be described by the equation:
\[ y = mx + b \]
where:
- \( y \) is the elevation of the gondola,
- \( m \) is the rate of change (slope),
- \( x \) is the time in minutes,
- \( b \) is the initial value (elevation at the start of descent).
In this case:
- The initial elevation \( b \) is 9,100 feet,
- The rate of descent \( m \) is -240 feet per minute (since the elevation is decreasing).
Thus, the equation representing the elevation of the gondola \( y \) after \( x \) minutes of descent can be written as:
\[ y = -240x + 9100 \]
This equation is linear and models the gondola’s descent effectively.