As part of a landscaping project, you put in a flower bed...

As part of a landscaping project, you put in a flower bed measuring 10 feet by 12 feet. You plan to surround the bed with a uniform border of low-growing plants. $$ \begin{array}{l}{\text { a. Write a polynomial that describes the area of the }} \\ {\text { uniform border that surrounds your flower bed. Hint: }} \\ {\text { The area of the border is the area of the large rectangle }} \\ {\text { shown in the figure minus the area of the flower bed. }} \\ {\text { b. The low growing plants surrounding the flower bed }} \\ {\text { require } 1 \text { square foot each when mature. If you have }} \\ {168 \text { of these plants, how wide a strip around the flower }} \\ {\text { bed should you prepare for the border? }}\end{array} $$

Transcript

0:00 All right.

00:02 So in this question, we have a landscaping project, and we have a flower bed that's 10 feet by 12 feet, and you're planning to build a border around it that's of uniform width.

00:13 So part a, you want to write a polynomial that describes the area of this said uniform border that surrounds your flower bed.

00:20 So here i drew it on the right side here, a diagram of the flower bed, which is in blue.

00:27 So it's 10 by 12.

00:28 Of course, it's not to scale, but, what matters are the numbers.

00:32 And here, the way you visualize it is basically you're growing your border uniformly on all sides.

00:39 You don't count the corners, really, because the corners are going to grow at a square or two times as fast as the vertical and horizontal components.

00:48 So basically, if you go horizontally and vertically, you're growing by an x feet.

00:55 You're trying to compute, well, what is the area of this border if my, if my, if my, uh, you know, width is x.

01:02 And so the way to compute that is to first realize that the total area of the flower bed and the border is going to be expressed as t of x, and that's going to be equal to your flower bed area plus your border area.

01:15 So if you know the total area and the flower bed area, you can just compete the border area by subtracting the two.

01:21 And so it's the total area and the flower bed area are easy to compute because they're rectangular.

01:25 So the total area is given by 10 plus 2x, which is the total vertical side here, and then 12 plus 2x, which is the little horizontal side here.

01:37 So i computed here, 2 of x equals 10 plus 2x times 12 plus 2x, and then you just expand it and then collect like terms.

01:44 Similarly, you do for the area of the flower bed, which is very simple, 10 times 12, which is 120...

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