Suppose that Gladys and George decided to use their tool...

Suppose that Gladys and George decided to use their tool shed as one side of the rectangular garden. This way they could use the 100 feet of fencing for three sides of the garden, (indicated by the bold lines in the figure below) and maybe they could have an even larger garden. What would be the dimensions of their new garden if they formed it this way andit were to have the largest possible area? Refer to the diagram. List any similarities with the problem from Session One. How many differences can you find? This is the tool shed This is the garden

Transcript

00:01 A farmer is to enclose a rectangular garden area by using 80 meters of fencing.

00:04 One of the sides of the garden will be against a barn.

00:06 Let l represent the length of the garden in meters and a represent the area of the garden in square meters.

00:12 First we want to write a formula that expresses a in terms of l.

00:16 So we'll let this side here be the barn.

00:21 And we have a rectangular garden area coming off of that barn.

00:26 So we're using 80 meters of fencing here to go around three sides.

00:32 L be the length.

00:34 I'm going to let this side here, so this side be the width.

00:37 We're going to get another expression for that here in a second because right now we know that the area of that rectangle just be the length of the width.

00:44 Well, we want this in terms of l only, not l and w, not the length and width.

00:49 So we know that the two of those lengths plus one width will equal the total 80 meters of fencing.

00:58 And we can solve this for w by subtracting 2l from b.

01:02 Both sides and we get w equals 80 minus 2 l so that's an expression we can use for the width the width is equal to 80 minus 2 l so then our area just becomes area equals our length times our width which we can say is 80 minus 2 l then we could use the distributed property there if we distribute that l that will give us 80 l minus 2 l squared.

01:35 Okay, so there's our area function.

01:37 So we want to find the maximum area of the garden.

01:40 So this area function is a quadratic function.

01:43 It has a negative leading coefficient, so that means it would open down like that, which means this has a maximum.

01:51 So to find where that maximum is, what value of l, since this would be the l and this would be the area, to find what value of l, that's maximized that.

02:02 That's where the tangent line is horizontal, so that's where the derivative of that function equals zero.

02:09 So let's find the derivative of that function.

02:11 If we find a prime, or the derivative of a, the derivative of 80l would just be 80, minus the derivative of 2l squared would just be 4l.

02:22 So we set that equal to 0.

02:25 So if we were to add 4l to both sides, we would get 4l equals 80...

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