Highway travel A state patrol station is located on a straight...

Question

Highway travel A state patrol station is located on a straight north-south freeway. A patrol car leaves the station at 9:00 A.M. heading north with position function s = f(t) that gives its location in miles t hours after 9:00 A.M. (see figure). Assume s is positive when the car is north of the patrol station. <IMAGE>
a. Determine the average velocity of the car during the first 45 minutes of the trip.

a. Determine the average velocity of the car during the first 45 minutes of the trip.

Accepted Answer

Hello there. Today we're going to 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 river boat begins its journey upstream at 7:30 a.m. Its position function S is equal to GFT, gives its location in kilometers. 2 hours after 7:30 a.m., if the river's current is flowing south and the boat's initial position is considered the origin, determine the average velocity of the boat during the first two hours of its journey. So that's our angle. Our angles we're ultimately trying to determine what the average velocity of this particular river boat is during its 1st 2 hours of its journey. So with that in mind, now that we know what we're trying to solve for, let's look at the graph that is provided to us by the problem itself. As you can see, our Y axis denotes the position in units of kilometers from the station, and our X axis denotes T, the time, and hours. And then we have a black curve denoting our function, or position function. Which is S equals G and T. So we have our curve, and as you can see the curve starts from the origin point. And then it steadily increases where till it peaks, and then it will decrease until it hits its lowest peak. Or one of its lowest peak, and then it will start to increase again, and then it'll decrease to its lowest peak, so it's the first low peak, but not the lowest peak, and then it'll start to tend upwards again. So it's shaped like a letter M of sorts. So, with that said, now that we have a visualization of what's happening in this particular prompt, we need to recall and note that the average velocity of the river boat during the first two hours of its journey can be given by the following formula, which states that the average velocity, which I'll call AV. is equal to the change. In position, which I'll call CIP change in position divided by change in time, which I'll call CIT. So the average velocity is equal to the change in position divided by the change in time. So we need to know and recall once again that the position function is given to us as S equals G of T. Where T once again represents the time and hours after 7:30 a.m. and we can express the average velocity over the first two hours as the average is equal to G 2 minus G of 0 divided by 2 minus 0, which is going to be equal to G of 2. Minus G of 0, and we need to divide everything by drum roll 2, let me simplify. So when we look at our graph, we will find that G of 0 represents the river's boat's position at T equals 0, which will be at 0 kilometers, and once again this is at 7:30 a.m. And G of 2 will represent the boat's position at T equals 2 hours, which is precisely 40 kilometers, which is at 9:30 a.m. So therefore, when we plug in our known variables, we will find that the average is going to be equal to, when we plug in our known variables since G2 is equal to 40 minus G0, which is equal to 0, divided by or by all divided by 2, so 40 divided by 2 is going to be equal to 20 kilometers per hour, and that is our final answer. Hooray, we did it. We solve for our final answer. So to quickly recap here, as you can see, so looking at our graph once again so we could visualize, I'm gonna circle in red are two points. So at G of 0. The boat's position is at T equals 0, which is 0 kilometers, so that's just simply at the origin point. And at G2 represents the position at T equals 2 hours, which is going to be at 40. Kilometers. So that's at the peak, our highest peak on our our curve. Fantastic. And that's it. So once again, our final answer, without a doubt has to be 20 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

Time Conversion

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