can i have some simple help with this rather long math problem?
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.
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A traveled 88 miles in 1 1/2 hours, so it is going 58 2/3 mi/hr
It will take another 99/(58 2/3) = 1.6875 hours to reach its destination
It had come 58 2/3 miles in the first hour, so it started out 187 + 58 2/3 = 242 2/3 miles from its destination.
So, after x hours, A's distance from its destination is
A = 242 2/3 - 58 2/3 x
I'm suspicious of these fractions, so check my math, as always.
Now do B, using the same idea.
Then you can answer the question.
To solve this problem, we will analyze the distances traveled by each truck at different times and use that information to determine the functions that describe their distances from the destination.
Let's start with Truck A:
- At 9 a.m., Truck A is 187 miles away from its destination, one hour after leaving from its origin.
- At 10:36 a.m., Truck A is 99 miles away from the destination.
To find the rate at which Truck A is traveling, we can subtract the distances traveled at different times and divide by the time difference:
Rate of Truck A = (187 miles - 99 miles) / (10:36 a.m. - 9 a.m.) = 88 miles / 1 hour 36 minutes
To convert 1 hour 36 minutes to decimal form, we use the conversion 1 hour = 60 minutes:
Rate of Truck A = 88 miles / (1 + 36/60) hours = 88 miles / 1.6 hours = 55 miles/hour
Now let's move on to Truck B:
- At 8:30 a.m., Truck B is 248 miles away from the destination.
- At 10 a.m., Truck B is 155 miles away from the destination.
Similarly, we'll calculate the rate of Truck B:
Rate of Truck B = (248 miles - 155 miles) / (10 a.m. - 8:30 a.m.) = 93 miles / 1.5 hours = 62 miles/hour
Based on the rates we calculated, the functions that describe the distances of the trucks from their destination over time can be written as:
Distance of Truck A = 187 - (time in hours after leaving) * 55
Distance of Truck B = 248 - (time in hours after leaving) * 62
Now, to determine which truck will arrive first, we can set the distances equal to zero and solve for the time when each truck reaches the destination:
For Truck A: 187 - 55t = 0
Solving for t, we find that t = 187 / 55 ≈ 3.4 hours
For Truck B: 248 - 62t = 0
Solving for t, we find that t = 248 / 62 = 4 hours
Therefore, Truck A will arrive at its destination first, after approximately 3.4 hours, while Truck B will arrive after 4 hours.