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A cyclist goes downhill for 15 min with a velocity to ling $v=40+64 t(\text { in } \mathrm{km} / \mathrm{h}),$ and then maintains the speed at the bottom for another 30 min. How far does the cyclist go in the 45 min?

Calculus 2 / BC

Chapter 26

Applications of Integration

Section 1

Applications of the Indefinite Integral

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Okay, so we have a cycler and their velocity is measured by the equation V of t equals 400 minus 20 t for the interval of zero. Um, 2 10 Uh, for time. So all the questions that we have are asking us how far that they have Britain, and we can't do that with the velocity. So we need to figure out the position equation. So the position function is found by finding the anti derivative of our distance. So assuming that's we are starting at zero. Because we start from the first point, we can see that the function s of T is equal to 400 t minus 20. T squared over to remember that when we are finding the anti derivative of exponents, we add one to the exponents and then we divide by that number, and then plus C s of zero equals zero equals 400 time zero minus 10. I'm zero squared plus C c equals zero. So our function is going to be s of T equals 400 t minus 10 t squared. So if we do that, we can figure out all the problems. So the 1st 1 asks this How far that she they have written at five minutes. So we put s of five equals 400 times five minus 10 times five squared, which is gonna be, but it's gonna be 2000 minus 10 times 25 which is 250. So they have would have ridden, uh, 1750 meters leaders over minutes and then before finding it at 10 minutes to do the same thing. But with 10 400 times 10 minus 10 10 squared, what gets 4000 minus 1000 that could weaken. See, that s of 10 is equal to 3000 meters. And the last question it asks us is how far the cyclists traveled when their velocity is 250 meters per minutes. So to figure that out, we're gonna plug in 250 back into our original velocity equation. So, for a religion of velocity equation, so we have V of t equals 400 minus 20 t. We're gonna plug in 250 into our results. We're gonna solve this. And at 7.5 minutes were at 250 um, or travelling at 250 meters per minute. And if we want to figure out the distance, we're gonna take 7.5 and plug it. Baggins or equation are this one. When, after some quick math, we gets 3000 minus 562.5. So we get 2000 300 are not there. 2000 437.5 meters. So you're three answers At five minutes it's 1750. At 10 minutes 3000. At 7.5 minutes, it's 2437.5 meters.

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