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.
i need help with the functions part. i already know truck A traveled 88 miles in 1 hour and 36 minutes and truck B traveled 93 miles in 1 hour and 30 minutes.
Using your calculated distances and times, the truck speeds are
A = 88/1.6 = 55 mi/hr
B = 93/1.5 = 62 mi/hr
Let x=0 correspond to 8:00 AM
Then we have points on the graphs of A and B as follows:
A: (1,187)
B: (0.5,248)
Then the equations of the graphs for the trucks are
A: y-187 = -55(x-1)
B: y-248 = -62(x-0.5)
or,
A; y=242-55x
B: y=279-62x
Now calculate the x-intercepts for these equations, and see which truck arrives at y=0 first.
Well, it seems like trucks A and B are racing to their destination! Let's break down the information given and find the functions to represent the distances traveled by each truck.
For Truck A:
We know that at 9 a.m., which is 1 hour after it started, it was 187 miles away from its destination. At 10:36 a.m., which is 1 hour and 36 minutes later, it was 99 miles away.
Let's find the rate at which Truck A is traveling first. If it traveled 88 miles in 1 hour and 36 minutes, we can convert that to hours by dividing by 60: 88 miles / (1 hour + 36 minutes/60) = 55 miles per hour (approximately).
Now we have the rate (r) for Truck A. We also know that at 1 hour after it started, it was 187 miles away from its destination. So, we can use the formula for distance (d) traveled as a function of time (t): d(t) = r * t.
In this case, d(t) = 55t + 187.
For Truck B:
At 6:30 a.m., Truck B started its journey, and at 8:30 a.m., it was 248 miles away from its destination. At 10 a.m., it was 155 miles away.
Similarly to find the rate, if it traveled 93 miles in 1 hour and 30 minutes, we can calculate the rate as 93 miles / (1 hour + 30 minutes/60) = 62 miles per hour (approximately).
Using the formula d(t) = r * t, where r is the rate and t is the time, the function for Truck B would be: d(t) = 62t + 248.
To determine which truck will arrive first to its destination, we can compare the distances traveled by each truck at the same time. Let's form an equation for when they both reach their destinations, d(tA) = d(tB).
55tA + 187 = 62tB + 248
Now we can solve this equation to find the time at which both trucks are at the same distance. Simply isolate one variable and calculate!
I hope that helps! Remember, even though trucks may be racing, we should always drive safely and keep the roads clown-free!
To find the function that represents the distance to the destination for each truck based on the time in hours after starting the trip, we can use the slope-intercept form of a linear equation: y = mx + b, where y is the distance to the destination, x is the time in hours, m is the rate of speed, and b is the initial distance from the destination.
Let's start with Truck A:
At 9 a.m., one hour after leaving from its origin, Truck A is 187 miles away from its destination.
At 10:36 a.m., Truck A is 99 miles away from the destination, which is 1 hour and 36 minutes (or 1.6 hours) after 9 a.m.
Using the given information, we can set up two data points to find the equation for Truck A's distance:
(1, 187) and (1.6, 99).
Using the slope formula, we can find the rate of speed (m) for Truck A:
m = (y₂ - y₁) / (x₂ - x₁)
m = (99 - 187) / (1.6 - 1)
m = -88 / 0.6
m ≈ -146.67
Using the point-slope form of a linear equation: y - y₁ = m(x - x₁), we can substitute one of the points and the calculated slope to find the equation for Truck A:
y - 187 = -146.67(x - 1)
y - 187 = -146.67x + 146.67
y = -146.67x + 333.67
So, the equation for Truck A is: Distance = -146.67(time) + 333.67.
Now, let's find the equation for Truck B:
Truck B started traveling to its destination at 6:30 a.m. (which is 0.5 hours before 7 a.m.).
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.
Using the given information, we can set up two data points to find the equation for Truck B's distance:
(0.5, 248) and (2, 155).
Using the slope formula, we can find the rate of speed (m) for Truck B:
m = (y₂ - y₁) / (x₂ - x₁)
m = (155 - 248) / (2 - 0.5)
m = -93 / 1.5
m = -62
Using the point-slope form of a linear equation: y - y₁ = m(x - x₁), we can substitute one of the points and the calculated slope to find the equation for Truck B:
y - 248 = -62(x - 0.5)
y - 248 = -62x + 31
y = -62x + 279
So, the equation for Truck B is: Distance = -62(time) + 279.
To determine which truck will arrive first to its destination, we need to find the time when each truck will reach 0 miles away from the destination. We can set the distance equation for each truck equal to 0 and solve for time:
For Truck A: -146.67(time) + 333.67 = 0
-146.67(time) = -333.67
time ≈ 2.28 hours
For Truck B: -62(time) + 279 = 0
-62(time) = -279
time ≈ 4.5 hours
Since Truck A will reach its destination in approximately 2.28 hours and Truck B will reach its destination in approximately 4.5 hours, Truck A will arrive first to its destination.
To find the functions that represent the distance traveled by each truck as a function of time, we can use the distance formula: distance = speed × time.
Let's start with Truck A:
- At 9 a.m., Truck A is 187 miles away from its destination, one hour after leaving the origin.
- At 10:36 a.m., Truck A is 99 miles away from the destination, which is 1 hour and 36 minutes (or 1.6 hours) later.
We can find the speed of Truck A by dividing the distance traveled (88 miles) by the time (1.6 hours):
Speed of Truck A = 88 miles / 1.6 hours = 55 miles/hour
Now we can use the speed to find the distance function for Truck A:
Distance of Truck A = Speed of Truck A × Time
Therefore, the function for Truck A is:
Distance of Truck A = 55t + 187
Where t represents the time in hours after starting the trip from the origin.
Now let's move on to Truck B:
- At 8:30 a.m., Truck B is 248 miles away from its destination, which is 2 hours after leaving the origin.
- At 10 a.m., Truck B is 155 miles away from the destination, which is 3.5 hours (or 3 1/2) later.
To find the speed of Truck B, we calculate the distance traveled (93 miles) divided by the time (3.5 hours):
Speed of Truck B = 93 miles / 3.5 hours = 26.57 miles/hour (rounded to two decimal places)
Using the speed, we can determine the distance function for Truck B:
Distance of Truck B = Speed of Truck B × Time
Hence, the function for Truck B is:
Distance of Truck B = 26.57t + 248
To determine which truck will arrive first at its destination, we need to find the time when the distance for each truck is equal. We can set the two distance functions equal to each other and solve for t:
55t + 187 = 26.57t + 248
Subtracting 26.57t from both sides, and then subtracting 187 from both sides, we get:
28.43t = 61
Dividing both sides by 28.43, we find:
t ≈ 2.15 hours
Therefore, Truck A will arrive at its destination before Truck B since it takes approximately 2.15 hours for Truck A to reach the destination, while Truck B will take longer.