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Problem 20 Medium Difficulty

A baseball diamond is a square with side $ 90 ft. $ A batter hits the ball and runs toward first base with a speed of $ 24 ft/s. $
(a) At what rate is his distance from second base decreasing when he is halfway to first base?
(b) At what rate is hits distance from third base increasing at the same moment?

Answer

a) -10.7
b) 10.7

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Video Transcript

mm six. We're told a baseball diamond is a square with side of 90 ft, Batter hits the ball and run towards first base with the speed of 24 ft/s. In part, they were asked at what rate is his distance from 2nd base decreasing when he is halfway to first base. Oh all right understand this question. First to draw a diagram. So I'll draw a square. This is our baseball diamond, wow. Season, wilson the show. Thanks. Hey, appreciate then we have between home and first base. This is our runner. Yeah. Read. Now I'll label the distance from the runner to first base of length. X. Do all the The length from the runner to 2nd base of Paul Y. Now we know the distance from first base. Second base is 90 ft and therefore by pythagorean theorem we know that 90 squared plus X squared equals widespread it we're also told that the rate which the runner is running towards first base the X. C. T. Well this is 24 ft per second and so get cT is negative 24 Feature 2nd because X. is decreasing as the runner gets closer to first base. You want to find the rate at which the distance from second base is decreasing. In other words. Dy DT Yeah. To find Dy DT I'll use implicit differentiation with respect to T. We have that two x times DX DT equals two Y times Dy DT and therefore Dy DT equals ready X over Y times the X. P. T. Yeah. Now because we're halfway to first face, please stop making. Mhm. It follows that X equals 45. This and therefore why is equal to be positive square root of 90 squared minus 40 plus 45 squared. How about this? Yeah, I'll say I don't say yes And this is 45 5. Want big big black ass and therefore follows that. Dy DT is equal to 45/45 route five times negative 24. That is it right. This is equal to -24 Routes five of the five. It's a right. Yeah. And this is in feet per second practice and yes. Yeah. Then in part B where as to what rate is his distance from 3rd base increasing at the same moment? Me. So this is a similar idea. Yeah. With thanks. Mhm. It's not being in a rough trade gates as practice Virginia. So once again, I draw a diagram. This is our baseball diamond watching you and we have our runner here halfway between home first. I'll call the distance. Yeah, Yeah. Yeah. From where he is the first base. I'll call this distance X. In the distance from where he is the 3rd base. Why? Now find the distance. You need to kind of know this distance here between 1st and 3rd base Well, because this is a square with side length of 90. This hypothesis here has a length of mhm 90 route to buy Pythagorean theorem person doing that. And therefore by pythagorean here and begin. Yes. Hello? Yeah, it or there's an alternative approach. Mom are up this hot cross instead. I love the distance at which the runner is getting further away from home. I'll call this X And the distance from 3rd base wise. And now we have a proper right triangle here. This is length 90. And therefore once again we have X squared plus 90 squared equals Y squared himself. Mhm. Yeah. Now in this case X is increasing because the runner is running towards first base. And so the expertise is positive. Yeah, 24 what? And again, well at this instinct were saying that X is equal to Half of 90, which is 45 and therefore, once again why is equal to 45? Route five? Also kids. Okay, Now, in order to find Dy DT, this is what we want to find. Once again, we have to use implicit differentiation. So we get two X. Dx DT equals two Y. Dy DT differentiate both sides by T and their course will be pretty. Y T T D Y T T equals great, X over Y times DX DT. Yeah. Closing in This is 45/45. Route five Times positive 24. And this is equal to positive 24 Ruth 5/5 and this is in feet per second. And so our answer is simply the opposite of our answer from part A Yeah.