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Consider the farmer in Exercise 7 whose field borders the river. Assume he wants the area of the plot to be 800 square feet. What should its dimensions be if he wants to use the least amount of fencing?

(a) $20 \times 40$(b) $20 \sqrt{2} \times 20 \sqrt{2}$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

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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.

02:40

A farmer has 1800 feet of …

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03:58

8A) A farmer has 3000 ft o…

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If the farmer in Exercise …

01:04

A farmer plans to fence a …

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01:32

Farmer plans to fence a re…

06:02

A dairy farmer plans to en…

01:16

A farmer with 4000 meters …

04:31

Minimum Length A farmer pl…

We're told that we want to consider the previous problem. Um should the dimensions be uh assume he wants the area of the plot to be um 800 square feet. What should its dimensions be if he wants to use his offensive. So again, we have a problem where we have, I should have done this one with the previous plans Have a river and the guy wants to put a fence around it, but he doesn't need to put a fence around the river, put a fence around the plot, but he doesn't need to put a fence along the river. So the area he wants is 800 square feet. And the perimeter then is is to A plus B. So we have solving for B. We have to a plus 800 over a. Taking the derivative of that. You get to -800 over a square substituting um finding are critical point. We set this equal to zero. So for a one a one is 20 and be one is 40. So he should have 40 ft here and 23 defense here in here. And so that would give him 800 Square feet with the minimum amount of fencing and that nana fencing is 880 ft. Now we have um I guess I probably should have just drawn the figure. Um since it's pretty simple here you have a plot where we have a rectangle. All right. And then we have on a partition at like that. Well, not necessarily even but three sanctions. So they tell us that the area of the total area is 18,000 square feet and they want us to minimize the amount of fencing so memorize. Yeah, well not the perimeter, but the amount of fencing total. So the amount of fencing is for A. This is A and this is B. So for A plus two B is the amount of fencing we have. So this for be substituted here and we get four A plus 36 360,000 over A. Take the derivative set A. One. So the critical point and we get four minus 360,000 over a one squared equals zero. So A one is 300 that means B one is 600 the total amount of fencing needed is 2020 400 ft. So we have I have 600 by 300. Give us the give us this area and minimum amount of fence needed.

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