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A farmer wishes to build a rectangular pig pen using as little fencing as possible. If the area of the pen is to be 225 square feet, what are its dimensions?

$$15 \times 15$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

Campbell University

Oregon State University

Baylor University

Idaho State University

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.

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we have a C. I'm gonna give these three problems together some pretty simple um And it's very related to one another. A farmer we are told has 102 125 square feet. Uh Farmer wishes to build a rectangular pig pen using as little fencing as possible if the area, the pig plan is 225 square feet. What is the dimensions? What are, what are the dimensions? What are uh So let's see here. We have, the area is 2 25 square feet. The perimeter is gonna be the two A plus to be would be A and B. Of the sides. Like sides. We can then solve this for A and substituted here. So we get two times the quantity to 25 over B plus B. Take the derivative of that. And we get um let's see here, I did I stick a minus sign in there. Obviously didn't mean to do that because I got the right answer. You should the minus, I shouldn't go in here. So taking the derivative, we get two times the derivative of this is minus 2 25 over squared through this is simply one. Of course. Now we set this equal to zero substitute and B one and solve that tells a breakthrough eligible problem and we get the one is 15 and a one is 15 also. So again we get a square and then the next problem so that the rectangle with a fixed area and minimum perimeter is a square. So it's basically the inverse problem of having a having a fixed perimeter and maximum area we have a fixed area and minimum perimeter. So we have and in fact, yeah, well they're obviously very related, so we have the area is eight times the the perimeter is to A plus to be solving this for a substituting in here, we get P equals two A. Over B. Plus to be taking the derivative, you get minus to a over B squared plus to set this equal to zero would be equal to our critical point. And we have um we wind up with the one equals the square root of A. Which then means that um a one is also the square root of a. Now we have, let's see here, what was the next one A farmer has built as a okay field bordering or straight river, you must design a rectangle plot. So you've got this river going here right here and he has a plot and he wants to put a fence around it. But in one case he isn't going to uh he doesn't need to put a fence along the border. And the other case he wants to put fence uh fence along the river, he doesn't need to put a fence along the river here. And in the other case he's gonna put fence along the river. Mhm. Yeah, so we can see here that he has, what did they say? 304 100 ft of fencing. So the perimeter in this first case is to A. These are A and this is B. So to a plus B. Is 400. And then area Cosby is the area, solve this for B. Plug it in here and we get the area is eight times 400 minus to a. Take the derivative expect A. And we get 400 minus four A. Set that equal to zero. So for the critical point A. One And we get a one is 100. So these would be 100 and this then would be 200. So he would have, what would this be? Uh Mhm 20,000 square feet enclosed. Now if he's going to put a fence along the river, now we have to include that in the perimeter, so this becomes to be going through everything. Um we get the area is then a times 200- A. Taking the river there. We get with respect to a we get 200 uh minus to a setting this equal to zero. And solving for a critical point A one a one is then 181 is 100. And obviously we should have known that from before because you know that, you know, this is a we shouldn't be a square. So now he only, he has basically half half the area as he did up here. So he's got to decide whether he wants to put a fence or needs to put a fence along the river

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