A train travels from city A to city B. The cities are 600 miles apart. The distance from city A at t hours
after the train leaves A is given by d(t) = 50t + t^2.
What is the average velocity of the train in miles per hour during the trip from A to B?
average velocity= distancegone/time=
= 600miles/time
600=50t+t^2
t^2+50t-600=0
(t+60)(t-10)=0
t= 10 hours (negative time is not allowed).
average velocity= 600miles/10 hrs=60mph
This is not calculus.
To find the average velocity of the train from city A to city B, we need to calculate the total distance traveled and the total time taken.
The distance traveled can be found by evaluating the function d(t) at t = 0 and t = t_b, where t_b is the time it takes for the train to reach city B.
Since the train starts from city A, the distance traveled at t = 0 is given by d(0) = 50(0) + 0^2 = 0 miles.
To find the time it takes for the train to reach city B, we can set d(t) equal to the distance between the two cities and solve for t:
600 = 50t + t^2
Now, we need to solve this quadratic equation. Rearranging it, we get:
t^2 + 50t - 600 = 0
This equation can be factored as:
(t + 60)(t - 10) = 0
So, t = -60 or t = 10. Since time cannot be negative, we discard t = -60. Therefore, it takes the train 10 hours to reach city B.
Now we know the distance traveled and the time taken. The total distance traveled is d(10) = 50(10) + 10^2 = 500 + 100 = 600 miles.
The total time taken is 10 hours.
To calculate the average velocity, we divide the total distance traveled by the total time taken:
Average velocity = Total distance / Total time
= 600 miles / 10 hours
= 60 miles per hour
Therefore, the average velocity of the train from city A to city B is 60 miles per hour.