1. The Garden Path

2. Story Problems
There are a number of related story problems that usually turn up when you are studying quadratic equations.
A gravel path of equal width is to be built around an 8’ by 8’ square garden. How wide can the path be if there is enough gravel for 80 sq’ ?

3. Story Problems
First, draw yourself a picture:

4. Story Problems
If the width of the garden is 8’ and the path is the same width “w” all the way around, then the width of the entire area is 8 + 2w
8 + 2w
8 + 2w

5. Story Problems 8 + 2w 8 + 2w 8 + 2w 8 + 2w
If the length of the garden is 8’ and the path is the same width “w” all the way around, then the length of the entire area is 8 + 2w
8 + 2w
8 + 2w
8 + 2w
8 + 2w

6. Story Problems 8 + 2w 8 + 2w 8 + 2w 8 + 2w
That means that the area of the garden plus the path is (8 + 2w)(8 + 2w)
8 + 2w
8 + 2w
8 + 2w
8 + 2w

7. Story Problems 8 + 2w 8 + 2w 8 + 2w 8 + 2w
Since the area of the garden is 8*8,
the area of the path will be (8 + 2w)(8 + 2w) – 64
8 + 2w
8 + 2w
8 + 2w
8 + 2w

8. Story Problems
They tell us that there is enough gravel for 80 sq’ so we know
(8 + 2w)(8 + 2w) – 64 = 80
64 +16w +16w +4w2 – 64 = 80
4w2 + 32w – 80 = 0
w2 + 8w – 20 = 0
(w + 10)(w – 2) = 0
w + 10 = 0 | w – 2 = 0
w = - 10 | w = 2
We can’t have a path that’s -10’ wide so the answer is 2’

9. Variations on a Theme Sometimes they tell us the dimensions of a room
and ask us something about a rug
placed so that there is an equally wide path around the rug.
In that case you may have to subtract 2w from the length and width of the room.
Draw yourself a picture.
Label it carefully.
Use your head!
Expect one answer that won’t work in real life.