Airline travel The following figure shows the position funct...

Question

Airline travel The following figure shows the position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 A.M. The plane returns to Seattle 8.5 hours later at 2:30 P.M. <IMAGE>
a. Calculate the average velocity of the airliner during the first 1.5 hours of the trip (0 ≤ t ≤ 1.5).

a. Calculate the average velocity of the airliner during the first 1.5 hours of the trip (0 ≤ t ≤ 1.5).

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A car starts from city A and travels towards city B. The position function of the car is given by S equals GFT, where S is the distance in kilometers from city A and T is the time and hour since departure at 8:00 a.m. The car makes a return trip to City A, arriving back 8.5 hours later at 4:30 p.m. Calculate the average velocity of the car during the 1st 2 hours of the trip, 0 is less than or equal to T and T is less than or equal to 2. Fantastic. So it appears for this particular problem, we're trying to calculate the average velocity of the car during the 1st 2 hours of this trip within the interval, parenthesis meaning open interval, 0 is less than or equal to T and T is less than or equal to 2. So with that in mind, now that we know what we're trying to solve for, we're trying to figure out what the average velocity of this car is during the 1st 2 hours of the trip. Let's look at our graph that is provided to us by the prom itself. The Y axis denotes our position. Which is in units of kilometers from city A, and our X-axis denotes the time T in units of hours. Now, as you can see, it appears to start our curve for our function starts at 00, the origin point, and increases until it hits a peak, and then it looks like it tends to be constant value, so it's neither increasing nor decreasing for a tiny bit. And then it starts to decrease rapidly and go down towards the touch our X axis, or in this case T-axis once again. So it's shaped like a mountain of sorts. So with that said, now that we have a good visualization of what's happening for this particular problem, we must first off, in order to solve this problem, we must recall and use the equation to help us solve for the average velocity of an object over a time interval. And as we should recall, we could write this equation as the average velocity, which I'll call AV is equal to the change in position, which I'll call CIP. So the average velocity is equal to the change in position, divided by the change in time, which I'll call CIT, so the change in time. So once again, the average velocity is equal to the change in position divided by the change in time, which is going to be equal to G of T2 minus G of T1, all divided by T2 minus T1, fantastic. So, therefore, at this point, we need to know and recall that we are asked to find the average velocity during the first two hours of this trip. So that means from T equals 0, which is at 8 a.m., And we're trying to find it from T equals 0 at 8 a.m. to T equals 2 hours. Awesome. So with that in mind, we need to note that at T equals 0, the car starts from city A, so that means that G of 0 is going to be equal to 0 kilometers, which, as you can see on our graph, when X is equal to 0, in this case it's T. So when T is equal to 0, That means that it's gonna be equal to 0, so when G of 0 is so when X is equal to 0, then it's gonna be 0 kilometers for the position. And similarly, for when T is equal to 2, that means the car's position is going to be G of 2. So if we look at our graph and we go to T2, so we look at T2, and we follow it up and up, you will find that the position is going to be equal to 125 kilometers. Fantastic. So, that means that we can calculate the average velocity now that we have found the variables that we need to plug into our equation. So that means when we plug in our known variables, we know that G of TF2. In this case, it's gonna be G of 2 minus G of 0, all divided by 2 minus 0, it's gonna be equal to 125 divided by 2. Which is going to be equal to, when we plug that into our calculator, that will mean that it's gonna be all equal to 62.5 when we round to one decial place, and it's going to be in units of kilometers per hour, and that is our final answer. Hooray, we did it. So once again, our final answer has to be 62.5 when we round to one decimal place, and its units of measurement are kilometers per hour. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Position Function

Average Velocity

Calculus of Motion

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