Bike race Theo and Sasha start at the same place on a straight...

Question

Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.

Theo: vT(t)=10, for t≥0

Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1

f. Suppose Sasha gives Theo a head start of 0.2 hr and the riders ride for 20 mi. Who wins the race?

Accepted Answer

Hi everyone, let's take a look at this practice problem, and this problem is dealing with a running competition. It says Jamie and Alex start at the same point on a straight track. Jamie runs at a constant speed of VJ, which is equal to 8 MPH. Alex's speed is V AFT, which is equal to 12T, or 0 is less than equal to T, which is less than equal to 1.5, and VAT, which is equal to 18, or T greater than 1.5, where T is in hours. Jamie is given a head start of 0.3 hours. If the race distance is 24 miles, who finishes first? The first thing we want to do is determine the completion time for Jamie. Since Jamie is running at a cons speed, this is easily calculated. So we'll call that completion time, TJ, and this is just going to be equal to the total distance of the race, which is 24 miles. Divided by their constant speed, which is 8 MPH, that means that Jamie's completion time is going to be equal to 3 hours. That means that Alex needs to complete the race in 2.7 hours in order to finish first, since Jamie was given a head start of 0.3 hours. So, next we need to calculate Alex's completion time. So, to begin with, we're gonna calculate the distance that Alex covers during the 1st 1.5 hours of the race. So, that distance X is going to be equal to the interval from 0 to 1.5 of the velocity function during this time interval, which is 12 T. and that gets multiplied by DT since we're going to be integrating with respect to T. So performing this integration, this is going to be equal to. I recall that when you integrate 12T with respect to T, you'll get 6 T squared. This needs to be evaluated from 0 to 1.5. So substituting in our limits of integration, this is going to be equal to 6, multiplied by the quantity of 1.5 squared minus 0 squared in quantity, and valuating this expression, this is going to be equal to 13.5 miles. So this is not a long, long enough distance in order to reach the end of the race, since the total race distance is 24 miles. So we need to determine the total distance covered since the beginning of the race, or a time T that is greater than 1.5 hours. So, we'll call that distance X of A, and so this is just going to be equal to the distance covered in the 1st 1.5 hours, which is the 13.5 miles. Plus, and for the second part here, we're just going to take their velocity function, which is a constant velocity after T greater than 1.5 hours, which is 18 MPH, and we need to multiply it by the quantity of T minus 1.5 in quantity, since that will give us the time that they have been running at 18 MPH after the 1.5 hours. Now, in order to cross the finish line, that means that X of A here needs to be equal to 24 miles, and we'll call the completion time T A. So, that means that 24 is going to be equal to 13.5. Plus 18, multiplied by the quantity of TA minus 1.5 in quantity. And so we need to solve this equation for TA. So, the first step is to subtract 13.5 from both sides of the equation. In doing so, we will have 10.5 is equal to 18, multiplied by the quantity of TA minus 1.5 in quantity, dividing both sides by 18, we'll get that 10.5 divided by 18. is equal to T A. Minus 1.5, then adding 1.5 to both sides of the equation, we see that TA is going to be equal to. 10.5 divided by 18, plus 1.5. Which is going to be equal to. 25 divided by 12, which is approximately equal to 2.08 hours. Now, this here is actually less than the required 2.7 hours to finish. First, that means that Alex. Finishes the race first, and that is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

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