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Problem 40 Hard Difficulty

A fence $ 8 ft $ tall runs parallel to a tall building at a distance of $ 4 ft $ from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

Answer

$\approx 16.65 \mathrm{ft}$

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Video Transcript

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