# A garden area is 30 ft long and 20 ft wide. A path of uniform width

is set around the edge. If the remaining garden area is 400 ft2, what is the width of

the path?

using the quadratic formula

I did the following:

Let x = distance between the inner rectangle and the outer boundry; i.e., x is the width of the border. Then

length is 20-x-x and the width is 20-x-x so (30-x-x)(20-x-x)=400

Solve for x.

Post your work if you get stuck.

## 5432

## To find the width of the path, we will use the quadratic formula to solve the equation (30 - 2x)(20 - 2x) = 400.

1. Start by rewriting the equation as a quadratic equation:

(30 - 2x)(20 - 2x) - 400 = 0

2. Expand the equation:

(30 - 2x)(20 - 2x) = 400

600 - 100x - 40x + 4x^2 = 400

4x^2 - 140x + 200 = 0

3. Apply the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In this equation, a = 4, b = -140, and c = 200.

4. Plug in the values into the quadratic formula and solve:

x = (-(-140) ± √((-140)^2 - 4 * 4 * 200)) / (2 * 4)

x = (140 ± √(19600 - 3200)) / 8

x = (140 ± √16400) / 8

x = (140 ± 128.06) / 8

5. Split the equation into two solutions:

x₁ = (140 + 128.06) / 8 = 268.06 / 8 = 33.51

x₂ = (140 - 128.06) / 8 = 11.94 / 8 = 1.49375

6. Since the width of the path cannot be negative, we discard x₂ as a solution.

Therefore, the width of the path is approximately 33.51 ft.

So, using the quadratic formula, we find that the width of the path is approximately 33.51 ft.