A farmer has 3600 ft of fencing and wants to fence off a...

A farmer has 3600 ft. of fencing and wants to fence off a rectangular field that borders a straight river. He does not need a fence along the river. What are the dimensions of the largest area that can be fenced? A) Experiment with the problem by drawing several diagrams illustrating the situation. Calculate the area of each configuration and use your results to estimate the dimensions of the largest possible field. (Round your answer to the nearest hundred feet) B) Find a function that models the area of the field in terms of one of its sides. C) Use your model to solve the problem and compare with your answer to part (a).

Transcript

00:01 In this problem, we're told that a farmer has 3 ,600 feet of fencing, and he wants to fence off a rectangular field that borders a straight river.

00:07 And we're told that he doesn't need to put fence along the river, because we can kind of use that as an edge.

00:12 We want to do the dimensions of the field that are lodgers that he can fence, meaning we want to find how we can maximize the area.

00:20 So in part a, what they want us to do is to draw diagrams to represent the situation.

00:26 Well, in this case, we're told that one of the sides is going to be a river, so this right here, will be the river, and then he's going to put a rectangular fence.

00:34 So it's going to be in the shape of rectangle.

00:36 Now, we're not told the length of the width.

00:38 So let's call this side l, which means these other two sides would be w.

00:42 And because it's a rectangle, i know that they're equal to each other.

00:45 So this would be a way that we could represent our diagram.

00:50 Now, in part b, what they want us to do is to find the function to model the area of the field in terms of one of the sides.

00:56 Well, keep in mind, we have 3 ,600 feet of fencing.

00:59 So that means when we add up these three sides, that should equal to 3600.

01:03 So it would have w plus w, which is 2w, plus that third side, which is l, this will all equal to 3600.

01:10 So now what i'm going to do is i'm going to solve this equation for l by subtracting 2w from each side.

01:16 So this leaves us with l equals 3 ,600 minus 2w.

01:20 So we can put that into our diagram.

01:25 So now that we have each of our sides in terms of w, what we can do is we can find the area.

01:30 Remember, to find the area of a rectangle, we multiply the length by the width.

01:34 Well, l we've already found in terms of w, would be 3 ,600 minus 2 times w, and then we're going to multiply this by the width, which is w.

01:42 So now, to get rid of the parentheses, we'll just go ahead and distribute, which means that our model will be 3 ,600 w minus 2w squared.

01:51 So this model would represent the area for the rectangle...

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