WEBVTT
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We're told that a fence 8 ft tall runs parallel
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to a tall building a distance of 4 ft from
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the building were asked what is the length of the
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shortest ladder that will reach from the ground over the
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fence to the wall of the building. So,
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yes, you're Yeah. Just might help to draw
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a picture here. So we have, On the
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one hand, we have a wall of the building
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. Then, on the other hand, we have
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our fence with the height of 8 ft. Yes
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, and we're gonna lean a ladder against this.
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Now, this is a distance of 4 ft from
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the wall, I should say, and we'll leave
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a ladder. It looks something like this, and
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the latter is going to have a length. Let's
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just call it El. And this length can actually
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broken up into two parts. I guess we'll call
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this, um, l one and little l two
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. Yes. Now, the question is, there's
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also an angle that the latter makes with the horizontal
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, which I'll call data. So what are L
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one and L two? In terms of data,
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we can transfer l data up here as well.
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What was it and mhm me. Now we know
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that l two over eight. This is equal to
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paper. Mm. Let me see. And,
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like, co sequence of data and so solving for
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l two, we have l two equals eight coast
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sequence data. Now, what is L. One
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? In terms of data, this is a little
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more difficult. So we have that l one over
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for the distance between the fence and the wall is
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equal to the sequence of data. So we have
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the l one equals four second of data and therefore
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the total length L as a function of theater is
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four second data plus eight coast sneak into data,
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actually, how to minimize. Yeah, l We're
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going to take the derivative and find critical values.
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So we have l prime of data. This is
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four times the derivative of seeking data, which is
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seeking data tangent Data, yes. Plus eight times
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the derivative of coast seeking data, which is negative
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. Coast seeking data, co tangent data so minus
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eight coast seeking data co tangent data. Yeah,
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I will set this equal to zero. And so
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we have that second data times. Tangent Data equals
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two co seeking data co tangent of theta. Let's
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write this in terms of signs and codes signs,
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if possible, maybe tangents. Yeah. So I
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multiply both sides, I Let's do sign and tangents
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. So I have this sign of data. We'll
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see times a tangent of data. Yeah, times
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the seeking to theta times The tangent of Fada again
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equals two. Yeah. Mm. So this is
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the same as tangent. Cubed of theta equals two
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. Yeah. Therefore, Tangent Data is two to
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the one third. Yeah, and so theta is
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equal to the arc Tangent of Cuba, relative to
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which is approximately 0.899 nine. And of course,
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this only makes sense because it's the only one within
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zero pi over two. Now we plug in Sata
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. Yeah, so we have that l of inverse
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tangent of the Cuban native to this is four times
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the second of the inverse tangent of Cuba routed to
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plus eight times the coast sequence of the inverse tangent
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of Cuba relative to. And if you either plug
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this into your calculator well, let's actually use triangles
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first. So we have the a triangle here with
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angle inverse tangent of Cuba to with so we have
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opposite side cube root of two adjacent side One sore
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hypotenuse is the square root of one plus two to
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the two thirds. And so the sequence this is
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four times this is one over the co sign.
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So this is hypotenuse square root of one plus two
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to the three halves or two to the two thirds
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over adjacent, one plus eight times the coast,
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seconds of our angle. So this is one over
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the sign. This is a high potty news over
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opposite, or the square root of one plus two
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to the two thirds over Cuba routed to and if
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you plug this into a calculator, approximately 16 65
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and the unit is in your feet.