Question
A car travels along the highway to Yuma at a steady speed. When it begins, it is 400 miles from Yuma. After 7 hours, it is 50 miles from Yuma. Which function describes the car's distance from Yuma? A. \( y=-10 x+400 \) B. \( y=350 x-400 \) C. \( y=-50 x+400 \) D. \( y=50 x+400 \)
Ask by Ayala French. in Saudi Arabia
Jan 13,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The car's distance from Yuma is described by the function \( y = -50x + 400 \).
Solution
To determine the function that describes the car's distance from Yuma over time, let's analyze the given information:
1. **Initial Distance**: At time \( t = 0 \) hours, the car is **400 miles** from Yuma.
2. **Distance After 7 Hours**: At time \( t = 7 \) hours, the car is **50 miles** from Yuma.
Assuming the car travels at a **constant speed**, the relationship between time \( x \) and distance \( y \) can be modeled by a linear equation:
\[ y = mx + b \]
Where:
- \( m \) is the slope (rate of change of distance with respect to time).
- \( b \) is the y-intercept (initial distance).
**Step 1: Determine the y-intercept (\( b \))**
At \( t = 0 \):
\[ y = 400 \]
So, \( b = 400 \).
**Step 2: Determine the slope (\( m \))**
Using the point \( (7, 50) \):
\[ 50 = m(7) + 400 \]
\[ m \times 7 = 50 - 400 \]
\[ m \times 7 = -350 \]
\[ m = -50 \]
**Step 3: Formulate the Equation**
Substituting \( m \) and \( b \) into the linear equation:
\[ y = -50x + 400 \]
**Conclusion**
The function that describes the car's distance from Yuma is:
\[
\boxed{y=-50x+400}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
The car starts 400 miles from Yuma and after 7 hours is only 50 miles away, which means it has traveled 350 miles during that time. The average speed can be calculated as 350 miles divided by 7 hours, resulting in 50 miles per hour as the rate of distance reduction. Thus, the function representing the car's distance from Yuma over time is \( y = -50x + 400 \). To double-check the logic, let's verify: If \( x \) is 7, substituting gives \( y = -50(7) + 400 = -350 + 400 = 50 \). This matches the problem's information perfectly, confirming that option C, \( y = -50x + 400 \), is indeed the correct function.
