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A man stars walking north at $ 4 ft/s $ from a point $ P. $ Five minutes later a woman starts walking south at $ 5 ft/s $ from a point $ 500 ft $ due east $ P. $ At what rate are the people moving apart $ 15 min $ after the woman starts walking?
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Calculus 1 / AB
Harvey Mudd College
University of Michigan - Ann Arbor
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.
A man starts walking north…
A man starts at a point $A…
Two people start from the …
take. We're told that a man starts walking north at four ft per second from a point p five minutes later, a woman starts walking south at five ft per second from a 50.500 ft due east of P. And were asked, At what rate are the people moving apart 15 minutes after the woman starts walking? Yeah, To understand this problem, let's draw a diagram. Zeffirelli. So we're just gonna put P r points at the origin? Yeah. Certain point. Move. Yes. Mhm. In the real stuff, I just was, like, 18 hours. But let me tell you some. Yes? Watch. Yes. Don't know how. Romeo. Yeah, right. Mhm. Okay, Now the man heads north ends up around here somewhere. The woman and head south ends up around here somewhere. The distance between horizontal distance from P to the woman is always 500 feet, right? All right, long hair just really should. And so there's a few different unknowns here. So we have this unknown this length here, which I'll call a this length here, which I'll call B and this last length here, which I'll call s sure make sure, actually, let's flip those. I'll call this a and this be It's Yeah, we should now from our diagram. Yeah. Mhm. You know, the man m has been walking for a total of oh, five miles? Yes. What b I did. I took, like because you needed to. Sorry, The man has been walking for a total of five minutes. Plus, then the 15 minutes after the woman starts for a total of 20 minutes. And so his distance from the point p a is equal to the rate he walks at, which is four ft per second, times the time in 20 minutes times the 60 seconds per minute. Yeah, which is equal to 4800. And the units is in feet course, A short 100. Because the history the woman w has been walking for a total of 15 minutes. See? So seriously? Mm, wow. And therefore, the distance be south of P. Yes. Mm hmm. The rate at which one walks five ft per second times the time. 15 minutes times 60 seconds per minute. This is equal to 4500. Right? And this is in feet. Yes. Now a plus B is one leg of the triangle mhm and we're told that d a d t. This is the radio. It's the man is walking, which is five ft per second. And the B D t is the rate at which the woman is walking, which is Yeah, Sorry. DDT was four ft per second and the B D t was five ft per second. It's around your people. Therefore, yes, as the the rate at which a plus B changes with respect to t. This is yeah, medal linearity of the derivative is D A d t plus DBT, which is four plus five, which is nine. And this is again in feet per second. Yeah, yeah. And at the moment in question, distance s between the man and woman. Well, this is the hypotenuse of a right triangle which is square root of a plus B at this time squared, plus the other leg which is 500 squared, which is equal to the square root of 4800 plus length of B which was 4500 squared plus 500 squared. And this is approximately 9313. But you were in the Nazi 0.43116 Yes. And this is in feet. This is a guy that loves me. Okay, well, I'm currently now. We want to find DS DT To do this, we're going to differentiate the Pythagorean theorem, So a plus B squared plus 500 squared equals s squared. And so we have differentiating both sides of respected T two times a plus B times d a d t plus DVD t plus zero is equal to two s times. DS DT mm. Never showed. Yeah, so we have DS DT is equal to a plus b times, and this is the derivative of a plus b t t all over s just like he's not. Yeah, and so plugging in our values. A plus b is 4800 plus 4500 times the derivative A plus B with respect to t. We found this was nine over s, which was approximately 9000 313 points. 43116 Plugging this into a calculator. This is approximately 8.987 Yeah, and the units are in feet per second. Read the entire who does she write for? I mean, once you, if you sign are in be having this conversation
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