Problem 79 Hard Difficulty

A high-speed bullet train accelerates and decelerates at the rate of $ 4\;ft/s^2 $. Its maximum cruising speed is $ 90\;mi/h $.
(a) What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 15 minutes?
(b) Suppose that the train starts from rest and must come to a complete stop in 15 minutes. What is the maximum distance it can travel under these conditions?
(c) Find the maximum time that the train takes to travel between two consecutive stations that are 45 miles apart.
(d) The trip from one station to the next takes 37.5 minutes. How far apart are the stations?

Answer

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WZ

Wen Z.

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Video Transcript

So we want to convert our cruising speed 2 ft per second. So if we have 90 MPH, we know that we can multiply that by the one hour over the 3600 seconds. And then we multiply the one mile, um and then on top is 5280 ft. So now what we get is we're going 132 ft per second. So at that conversion, um, we want to do the minutes two seconds, so we know it's 15 minutes and in seconds it's going to be 900 seconds. So with that accelerating from rest, we have that after equals 4 ft per second squared, the F T is going to give us for t plus C. But since V of zero equals zero, we note that via zero will equal zero plus c. So it's just going to be zero. So R V of T function is actually just 14. Now we want to take the anti derivative again to get our fft. Then that's gonna give us 4/2, which is just to so to t squared, plus some other value C two. We'll call it, and at it is at zero when t equals zero. So that means that s of zero equals C two, which equals zero. So our actual position function is just going to be two t squared, plus, actually just to t squared. Then it's time to get our max cruising speed. So we get that four t equals 132 because that was our feet per second. That's t equals 33 seconds. Then putting that into our position function s of 33. That tells us that the distance traveled as 2178 ft. Then we multiply this time at cruising speed by the speed to get the remaining distance traveled. So it's going to be 900 seconds. Times are 132 ft per second, and that ends up giving us 118,000 800 feet. So the total distance traveled is going to be this value plus this value. And that's gonna give us a final answer of, um, 120,978 ft, which we will convert back into miles. That's 2 22 9125 miles. All right now moving on to Part B. Um, we found some important values in part A. So now we want to determine the time traveled at cruising speed. So if we take the 900 seconds minus the two times 33 um, so that's the two coming from are other equation. So now what we have is we subtract this and now we get 834 seconds, so the distance traveled when accelerating and accelerating is going to be two times 21 78 which is going to give us for 356 feet if we multiply the time at cruising speed. Um, now what we have is our 834 seconds times the 132 ft per second that's going to give us 11, actually 110,088. He So that means that the total distance traveled is going to be this value plus this value converted into miles, which is 21 point 675 miles for part C. We use our previous information. We know that the total distance to travel is, um 2270. If we take the 45 miles um So our total distance travel over 45 miles is 237,600 ft. So at cruising speed, that's going to be the 200 37 600 minus the 4 356 which is the amount of time accelerating, accelerating and decelerating. So it's 273,600 actually 200 33,244 ft. Then we use the equation for constant speed, which is time equals distance over speed. So the time that we end up getting is 1767 seconds. Converting that into minutes. We know that that is 35 5 minutes. Then we have one last part to do part. Dean, we know that the total time to travel, um, if the time required to accelerate and decelerate is 66 the try out the car, the train travels 4356 ft. If we have 30. Uh huh. We have 30 seven 0.5 minutes. That is going to take 2250 seconds. Um, and then the cruising speed time would be 20 to 50 minus 66 because that's are accelerating and decelerating time. So that's 2184. Then, using our equation of constant speed, we get distance equals the 132 ft per second times our 2184 seconds. So that means our distance is going to be 288,000 288 ft. Convert that back into miles, and what we end up getting is 55.4 to 5 miles. That is our final answer for party.