Two freight trucks are traveling to the same destination, and each are traveling at a constant speed. Truck A is 187 miles away from its destination at 9 a.m., one hour after leaving from its origin, and is 99 miles away from the destination at 10:36 a.m. Truck B started traveling to its destination at 6:30 a.m. Truck B is 248 miles away at 8:30 a.m. and 155 miles away at 10 a.m. Analyze each situation to determine a function that finds the distance to the destination for each truck based on the time in hours after starting the trip from the origin to the destination. Then use the functions to determine which truck will arrive first to its destination.(4 points)
To determine the functions that represent the distance to the destination for each truck based on the time in hours after starting the trip, we can use the information provided.
For truck A, we know that at 9 a.m., one hour after leaving from its origin, it is 187 miles away from its destination. At 10:36 a.m., it is 99 miles away from the destination.
Let's define the function for truck A as D_A(t), where D_A(t) represents the distance to the destination for truck A at time t hours after starting the trip.
We can use the slope-intercept form of a linear equation to find the equation of the line passing through the points (1, 187) and (1.6, 99), where 1 represents one hour after starting the trip (9 a.m.) and 1.6 represents 1 hour and 36 minutes after starting the trip (10:36 a.m.).
The equation of the line can be written as:
D_A(t) = mt + b
where m represents the slope and b represents the y-intercept.
First, let's calculate the slope:
m = (99 - 187) / (1.6 - 1)
= -88 / 0.6
= -440 / 15
Now, let's substitute one of the points (1, 187) into the equation to calculate the y-intercept:
187 = (-440 / 15)(1) + b
Solving for b, we have:
b = 187 + (440 / 15)
= 1120 / 15 + 440 / 15
= 1560 / 15
= 104
Therefore, the function that represents the distance to the destination for truck A based on the time in hours after starting the trip is:
D_A(t) = (-440 / 15)t + 104
Now let's analyze truck B. We know that truck B started traveling to its destination at 6:30 a.m. At 8:30 a.m., it is 248 miles away, and at 10 a.m., it is 155 miles away.
Let's define the function for truck B as D_B(t), where D_B(t) represents the distance to the destination for truck B at time t hours after starting the trip.
Again, we can use the slope-intercept form of a linear equation to find the equation of the line passing through the points (2.5, 248) and (4, 155), where 2.5 represents 2.5 hours after starting the trip (8:30 a.m.) and 4 represents 4 hours after starting the trip (10 a.m.).
The equation of the line can be written as:
D_B(t) = mt + b
where m represents the slope and b represents the y-intercept.
First, let's calculate the slope:
m = (155 - 248) / (4 - 2.5)
= -93 / 1.5
= -62
Now, let's substitute one of the points (2.5, 248) into the equation to calculate the y-intercept:
248 = (-62)(2.5) + b
Solving for b, we have:
b = 248 + (62)(2.5)
= 248 + 155
= 403
Therefore, the function that represents the distance to the destination for truck B based on the time in hours after starting the trip is:
D_B(t) = (-62)t + 403
Now, we can determine which truck will arrive first to its destination by comparing the functions D_A(t) and D_B(t).
To analyze this, we need to find the time t when the distance to the destination for truck A (D_A(t)) equals the distance to the destination for truck B (D_B(t)).
(-440 / 15)t + 104 = (-62)t + 403
Let's solve this equation:
(-440 / 15 + 62)t = 403 - 104
(-440 / 15 + 62)t = 299
Now, we can solve for t:
(-440 / 15 + 62)t = 299
(-440 + 930)t = 8985
490t = 8985
t = 8985 / 490
t ≈ 18.34
So, truck A will arrive first to its destination in approximately 18.34 hours after starting the trip.