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A low stone wall 27 inches high is 64 inches away from a building. What is the length of the shortest pole that passes over the fence and reaches the building? (Hint: The length of the pole is composed of two portions, the portion from the ground to the fence, and from the fence to the building. See Figure $13 .$ Use the theorem of Pythagoras along with similar triangles to find the total length.)

$$125 \mathrm{ft}$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

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University of Michigan - Ann Arbor

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Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

10:34

The 8-ft wall shown here s…

08:13

A fence $ 8 ft $ tall runs…

11:25

Shortest beam The 8 -ft wa…

09:41

02:52

A fence $8 \mathrm{ft}$ ta…

08:41

A fence 8 ft tall runs par…

Yeah, it's kind of an interesting problem that I don't think I've seen something like this before. Um but kind of a fun problem a little bit um have to think about a little bit. So we have this wall here. So actually let me actually make this kind of a red wall here and then we have a building over here. So we have this wall protecting our building and we want to find out what is the minimum ladder length that we could get to uh basically get over the wall to this, to the building. So we're told that this wall is 27 inches high and it is 64 inches inches from the from here. So notice these are all these these are nice. Um There's nice uh factors of three here, so that hopefully this this is gonna work out fairly nice. You know, these are just arbitrary values, we'll probably get a mess. So what I'm gonna do is I'm going to say, okay we have this we have basically this triangle here and we have this triangle and we have this triangle. And so we know the length of this ladder is going to be this length here, plus this length here. And this length is um this length squared is 27 square plus a square. And this length here is 64 squared plus B squared. So we have these two expressions for the squares of the length and we know the total length is um l one plus two. And then they're using similar triangles, which they suggested we know that A oh my Divided by 27 has to be uh 64 divided by B. Okay so the races of these sides have because these are similar triangles. And this is a similar triangle to we could have used that similar triangle but it would have been lucky. So we can use this and this is similar triangles, they're not equal but they have the same angles, right? So they scale like you know like this the ratio of their sides. So we have this constraint and that basically is a constraint that says that basically says that this point needs to be you know on on the ladder. Okay. It's obviously a larger letter would work, right? But we need a constraint to say that you know now with that we can use this to solve for software A. And and plug that into L. And using these. So we can L. Equals as a function of B. Is this kind of mess here? Um But we can take a derivative of it. It's not pretty but we can do it and so we get L. Prime are optimal solution is uh this I guess I'm not gonna talk it out but it's a pretty messy thing but we can do some simplifying on this and we'll find out that in fact we have actually a very nice solution and that is the one who was 48 inches or four ft. So that's we're gonna browned up being four ft above this wall. Um and then 81 is 36 inches, so we're going to be three ft um from here. So we need and then if we look at what L one plus L. Two is, you know, substitute these back into here, we get that the latter is 125 inches, so we have to put it down three ft from the wall and it's 125 inches. So what's gonna wind up hitting the building? Um four ft above above the height of the wall? Mhm.

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