To solve this problem, we need to determine the function that describes the car's distance (y) from Yuma in terms of the time (x) it has been traveling.
Here's a step-by-step solution:
1. Initial Distance: When the car starts, it is 400 miles away from Yuma. This is given, so our initial condition is:
[tex]\[
y(0) = 400
\][/tex]
2. Distance After 7 Hours: It is given that after 7 hours, the car is 50 miles from Yuma. This provides us with the second condition:
[tex]\[
y(7) = 50
\][/tex]
3. Determine the Speed: To find the car's speed, we can use the distances at two different times. Initially, the distance was 400 miles, and after 7 hours, it decreased to 50 miles. The distance change over time gives us the speed:
[tex]\[
\text{Distance change} = 400 - 50 = 350 \text{ miles}
\][/tex]
[tex]\[
\text{Time change} = 7 \text{ hours}
\][/tex]
[tex]\[
\text{Speed (steady)} = \frac{\text{Distance change}}{\text{Time change}} = \frac{350 \text{ miles}}{7 \text{ hours}} = 50 \text{ miles per hour}
\][/tex]
4. Formulating the Function: The distance from Yuma is decreasing at a steady speed as time increases. Thus, we can describe this relationship with a linear function where the slope represents the speed (negative because the distance is decreasing), and the y-intercept represents the initial distance.
Therefore, the function should be:
[tex]\[
y = \text{initial distance} - (\text{speed} \times \text{time})
\][/tex]
Substituting the initial distance (400 miles) and the speed (50 miles per hour):
[tex]\[
y = 400 - 50x
\][/tex]
5. Simplified Function: The function that describes the car's distance from Yuma in terms of the time is:
[tex]\[
y = -50x + 400
\][/tex]
Given the choices:
A. [tex]\( y = 50x + 400 \)[/tex]
B. [tex]\( y = -50x + 400 \)[/tex]
C. [tex]\( y = -10x + 400 \)[/tex]
D. [tex]\( y = 350x - 400 \)[/tex]
The correct function is [tex]\( y = -50x + 400 \)[/tex].
So, the answer is:
B. [tex]\( y = -50x + 400 \)[/tex]
