A fence h feet high runs parallel to a tall building and w feet from it (Figure 23 ). Find the l

step 1 So here we have a fence that is H feet high. So let's say this here is our fence and it's H feet

A fence h feet high runs parallel to a tall building and w...

Key Concepts

Reflection Principle

This principle involves reflecting one of the endpoints across a barrier to transform a segmented path into a straight-line distance. In geometric optimization problems, such as finding the shortest ladder length that crosses a barrier, the reflection method simplifies the problem and helps identify the minimal distance path.

Pythagorean Theorem

The Pythagorean Theorem is fundamental in relating the horizontal and vertical components of right triangles. In this context, it is used to connect the ladder's length with its horizontal distance from the building and its vertical height, forming the basis for the distance calculations required in the optimization process.

Calculus Optimization

Calculus optimization involves setting up a function that models the length of the ladder and then finding its minimum value by calculating derivatives. This technique is applicable in problems where a varying quantity, subject to certain constraints, must be optimized to achieve a minimum or maximum value.

Similar Triangles

Similar triangles help in establishing proportional relationships between different segments within the geometric configuration. They are crucial in expressing the relationships between the dimensions of the fence, the distances from the building, and the corresponding segments of the ladder, thereby enabling the formulation of equations that lead to determining the optimal ladder length.

Transcript

00:01 So here we have a fence that is h feet high.

00:05 So let's say this here is our fence and it's h feet.

00:09 And it runs parallel to a tall building and w feet away from it.

00:14 So we're assuming that this building is infinitely tall and that it is w feet away from the fence.

00:22 So if i draw a line here, this here is w.

00:28 So what we want to do is we want to find the length of the shortest ladder, that will reach from the ground across the top of the fence to the wall of the building.

00:38 Okay, so such a ladder could be something like this.

00:43 All right, so it's touching the top of the fence and then it touches the ground.

00:48 Okay, so we'll call this ladder length out.

00:53 Okay, so as you can see, you can shift the ladder up the building, so it'll be more steep, or you can shift it down, and then it'll be less steep.

01:02 So the idea is we want to find the ladder with the shortest length.

01:07 Okay, so we're going to introduce another variable here, and that is going to be the angle theta here, which is the angle that the ladder makes with the ground.

01:19 Make it a little bit bigger, so this angle here is theta.

01:23 Okay, and then i'm also going to draw a line here to connect the ground from the fence to the ladder.

01:32 Okay, so there are a couple ways of doing this, but i think that the easiest and the cleanest way to do this is to actually split this ladder up into two portions.

01:44 Okay, so this entire length is length l.

01:47 But we're going to call this length here.

01:50 So from the ground to the fence, we're going to call that length l1.

01:56 So meaning the first part of the ladder.

01:58 And then the second part of the ladder, which runs from the fence to the building, we're going to call.

02:03 Call that l2.

02:06 So clearly, l is going to be l1 plus l2.

02:09 Right, so let's write that down.

02:11 The length of the latter is the first part of the ladder plus the second part of the ladder.

02:16 So what do we want to do this? well, if we do this, then we can actually express l1 and l2 in terms of theta, h, and w.

02:25 Okay, so let's start with l1 first.

02:29 So l1 here is the hypotenuse of this right angle triangle.

02:34 Okay.

02:35 So if we have our angle theta here, then that means sign theta is going to be h over l1.

02:42 Okay, so let's write that on the side, which actually will write underneath.

02:47 So sine theta is opposite over hypotenuse, which is h over l1.

02:52 And this means l1 is equal to h over sine theta.

02:59 And then if we look at l2, we can actually draw a triangle like, this.

03:06 So you would agree with me that this angle here is also theta, right? and then this, this dotted green line that we just drew is w.

03:16 So that means cos theta is going to be equal to adjacent over hypotenuse.

03:23 So that's adjacent, which is w and the hypotenuse in this case is l2.

03:29 Okay, so which means l2 is equal to w over coast theta.

03:36 And now we can go ahead and put l1 and l2 back into our equation for l.

03:41 So this is going to be equal to h over sine theta plus w over coast theta.

03:51 So h and w are just constant, and now theta is our only variable.

03:56 Okay, so now we can go ahead and find critical values by taking the derivative of l with respect to theta.

04:02 So this is l prime of theta, and that's going to be equal to negative h, and that'll be, sign theta to the power of negative 2 and then chain rule so we need a coast theta and this is going to be minus w and this will be coast theta to the power of negative 2 and then chain rule says we need a negative sign theta so that just becomes a plus and then we multiply by sign theta okay so let's reorganize this a little bit we'll pull the positive one to the front so this is really w sign theta let's write that a little bit bigger.

04:43 W.

04:44 Sine theta over cos squared theta minus h -cose -theta over sine -squared theta.

04:57 So one thing is we have to realize what our domain for theta is.

05:03 So theta, in this case, has to be greater than zero.

05:08 So imagine if this ladder was super, super, super flat...

Please give Ace some feedback