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)
Let's first find the equation for the distance to the destination for truck A.
From 9 a.m. to 10:36 a.m., there is a time difference of 1 hour and 36 minutes, which is equivalent to 1.6 hours.
We can use the formula for distance traveled at a constant speed:
distance = speed x time
Let's let x represent the time in hours after leaving the origin.
At 9 a.m. (x = 1), truck A is 187 miles away from the destination. At 10:36 a.m. (x = 2.6), truck A is 99 miles away from the destination.
Using these two points, we can set up a system of equations:
187 = speed x 1
99 = speed x 2.6
Simplifying the equations:
speed = 187 (equation 1)
99 = 2.6 x speed
Substituting the value of speed from equation 1 into equation 2:
99 = 2.6 x 187
Solving for speed:
speed = 99 / 2.6 = 38.077
Therefore, the equation for the distance to the destination for truck A is:
distance_A = 38.077x
Now let's find the equation for the distance to the destination for truck B.
From 6:30 a.m. to 8:30 a.m., there is a time difference of 2 hours. From 8:30 a.m. to 10 a.m., there is a time difference of 1.5 hours.
At 6:30 a.m. (x = 0), truck B is 248 miles away from the destination. At 8:30 a.m. (x = 2), truck B is 155 miles away from the destination. At 10 a.m. (x = 3.5), truck B is 0 miles away from the destination.
Using these three points, we can set up a system of equations:
248 = speed x 0
155 = speed x 2
0 = speed x 3.5
Simplifying the equations:
speed = 248 (equation 1)
155 = 2 x speed
0 = 3.5 x speed
Substituting the value of speed from equation 1 into equations 2 and 3:
155 = 2 x 248
0 = 3.5 x 248
Solving for speed:
speed = 155 / 2 = 77.5
speed = 0 / 3.5 = 0
Therefore, the equation for the distance to the destination for truck B is:
distance_B = 77.5x for 0 <= x <= 2
distance_B = 0 for x > 2
To determine which truck will arrive first, we can compare the time it takes for each truck to reach the destination.
For truck A, we can set the equation distance_A = 187 - 38.077x = 0 and solve for x:
187 - 38.077x = 0
38.077x = 187
x = 187 / 38.077 = 4.908
For truck B, we know that truck B reaches the destination at x = 2. Therefore, truck B will arrive first.
Truck B will arrive first to its destination.