WEBVTT
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So first we want X to be the distance from
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B to C so the distance from the island to
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see is going to be the square root of X
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squared plus 25 because five squared is 25 and the
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distance from C to D. Um well, right
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. Like this, C D is going to be
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13 minus X, so the total distance travel will
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be the square root of X squared plus 25 plus
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13 minus x. So if K is the energy
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per kilometer it takes to fly over the land,
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then our total energy, which will be modeled by
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the function F X, is going to be 1.4
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k times X squared plus 25 plus K times 13
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minus x. Then we find e prime of X
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. And keep in mind that K is a constant
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and what we end up getting as a result is
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K Times 1.4 x minus square root of X squared
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minus 25 over the square root of X squared plus
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25 as plus 25 in both cases. And then
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we want to set that derivative equal zero. Now
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we always remember that the bottom doesn't really matter.
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We're more focused on the top because when the top
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equals zero, the numerator is equal zero. Um
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, the whole fraction equals zero. So we have
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since this constant could just be divided out. What
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we really have is 1.4 x minus the square root
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of X squared plus 25. We solve for X
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, and we end up getting that X is equal
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to plus or minus the square root of 25 over
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, um 196 but an extra 4.96 section. But
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in actuality, since it's obviously just going to be
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the plus answer, what we have is 51 kilometers
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. Then we see that a prime of one gives
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us a negative 07 k. So that's obviously less
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than zero, and then the prime of 10.
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So we're picking a point above and below this 51
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we see that this one right here is 0 to
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5 k, which is greater than zero. So
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what we have is we are decreasing in slope,
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and then we reach this 5.1 kilometers and then we're
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increasing in slope. So what that tells us.
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Is that 5.1? Um, this is a minimum
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. Then we have our next part, part B
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. So we know that if w over l is
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large, then the bird would fly to a point
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C that is closer to be than two D in
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order to minimize the energy used flying over the water
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. However, if it's small, then the bird
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would fly to point C that is closer to D
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than to be in order to minimize the distance of
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the flight. So what we have is the e
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equals w times X squared plus 25 plus L Times
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13 minus x Then a prime of X is going
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to give us X w fax over squared of X
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square plus 25 minus l vax. When we isolate
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, we set this equal zero And when we isolate
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W over l, we end up getting that.
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This is X squared plus 25 over X so the
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above ratio will minimize the energy. If the bird
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aims for the point that is X kilometers away from
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beat. For part C, going directly to D
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means that X will equal 13, so w over
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l is going to equal the square root of 13
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squared, which is 169 plus 25 over 13.
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And that's gonna give us 107 Um, so there
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is no value of w over l for which the
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birds should go directly to be, and then our
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last problem, the if the birds choose a path
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path that minimizes energy And that's gonna be e D
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X, which equals 14 K x over Route 25
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plus X squared minus K. We set that equal
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to zero, and what we end up finding is
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that 14 k x equals K times X, the
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square root of X squared plus 25 case cancel.
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We solve for X. We get that, um
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14 k. If we let this equal see and
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then x equal four and K equal one we get
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The sea is approximately 1.6