WEBVTT
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so we know that we have a runner and he's
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sprinting around a circular track at a constant speed of
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7 m per second. So we have a person
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right here running around at 7 m per second.
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Then another thing we want to keep mind is the
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fact that this, uh, has a radius of
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100 m. So this is also going to be
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100 m. Then we have their runners friend standing
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at a different distance of 200 m from the center
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. So then we have another 100 right here and
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the friend, and we want to know how fast
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is the distance between the friend changing when the distance
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between them is 200 m. So we want to
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know that when this is the case right here,
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when this is 200 m, what is that rate
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of change? And we have an angle right here
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and that's gonna become important. So with this in
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mind, we want to draw this diagram that'll help
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us somewhat. Um, then we want Thio relate
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our quantities using the coastline law because we have a
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lot of signs and one angle. So we see
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that C squared, which is this C is going
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to be the distance. So C squared is equal
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to a squared plus B squared minus two a.
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B Co Cynthia So using the law of coastlines in
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differentiating both sides, we get that to see d
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c d t equals to a B signed theater.
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Do you say the D team solving for a D
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. C D. T. Uh, in order
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to do that, we need to find science,
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data and data DT eso because it's not given to
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us, we're going to have to find it ourselves
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. So since the runner is moving at 7 m
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per second, we want to relate that to the
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arc length formula, which tells us that the radius
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times data is equal to the length. So differentiating
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both sides again, we'll get that d r d
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t feta. And this is chain role plus r
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d theta DT is equal. Teoh de l d
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T Onley rearrange our values and plug in the ones
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that we know. We know that d l D
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t is seven over 100 so we have seven over
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100 then we know that that is equal to um
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, de data over DT. So that's 0.7 radiance
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per second. So with this in mind, we
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now need to find data. So we'll go back
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to our coastline law. I'm in plugging important values
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, so we'll have. The coastline of Fada is
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equal to C squared minus B squared, minus a
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squared over to a B. And when we plug
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in the values that we know for A, B
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and C based on our diagram up here, we
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end up getting at the coastline of Fada is 1/4
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um and then we could solve for theta. Or
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we could use the Pythagorean identities, which tells us
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that the sine squared of theta plus the co sine
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squared of theta is equal to one. So just
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solving this way we get that the sign of data
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is equal to Route 15/4. So with all these
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values, now we go back what we had originally
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, which is when we solve for DC DT and
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what we end up getting is that this is gonna
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be a b sine theta d theta DT all over
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. See playing these values that we've already solved for
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and that were given We get that D. C
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D. T is seven Route 15 over four,
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which means that when the distance between the runner and
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his friend is 200 m, they're changing their changing
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at a distance of seven times the route 15/4 meters
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per second.