To determine which equation best represents the floor number of an elevator ascending at a steady rate, we need to understand a few key concepts about the context of the problem and the nature of linear equations.
We have four equations:
A. [tex]\( y = 2x - 75 \)[/tex]
B. [tex]\( y = -2x + 75 \)[/tex]
C. [tex]\( y = 2x + 75 \)[/tex]
D. [tex]\( y = -2x - 75 \)[/tex]
Here's a detailed analysis of each one:
### Equation A: [tex]\( y = 2x - 75 \)[/tex]
- This equation has a positive slope of 2. A positive slope suggests an upward or ascending trend, which is what we are looking for in an elevator ascending.
- However, the y-intercept is -75. This implies that at time [tex]\(x = 0\)[/tex], the elevator would be at floor -75, which is an unrealistic starting point for an elevator.
### Equation B: [tex]\( y = -2x + 75 \)[/tex]
- This equation has a negative slope of -2. A negative slope indicates a downward or descending trend, meaning the elevator would be going down rather than up.
- Although the y-intercept is positive, indicating it starts at floor 75 at time [tex]\(x = 0\)[/tex], the negative slope disqualifies it from modeling an ascending elevator.
### Equation C: [tex]\( y = 2x + 75 \)[/tex]
- This equation has a positive slope of 2. Like Equation A, a positive slope indicates the elevator is ascending.
- The y-intercept is +75. This means that at time [tex]\(x = 0\)[/tex], the elevator is at floor 75, which is a logical and realistic starting point for an elevator in a building.
### Equation D: [tex]\( y = -2x - 75 \)[/tex]
- This equation has a negative slope of -2. Similar to Equation B, a negative slope indicates a descending elevator.
- The y-intercept is also -75, which further indicates an unrealistic starting floor when [tex]\(x = 0\)[/tex].
### Conclusion:
Among the given options, Equation C [tex]\((y = 2x + 75)\)[/tex] has a positive slope, indicating an upward or ascending motion, and starts at a positive and realistic floor number (75) at time [tex]\(x = 0\)[/tex]. Thus, it best represents an elevator ascending at a steady rate.
So, the correct answer is:
C. [tex]\( y = 2x + 75 \)[/tex]
