a farmer fences off a small.rectangular plot of land thay is 60 feet by 100 feet. within that area, a garden is planted and surrounded by a walkway of uniform width. how wide is the walkway if the area of the planted area is one half of the area of the plot?

let the width of the pathway be x ft

width of garden = 60-2x , x < 30
length of garden = 100-2x ,

area of garden = (60-2x)(100-2x)
= 6000 - 320x + 4x^2

area of plot = 6000
half of plot = 3000

4x^2 - 320x + 6000 = 3000
x^2 - 80x + 750 = 0
I will complete the square:
x^2 - 80x + 1600 = -750 + 1600
(x-40)^2 = 850
x-40 = ±√850 = ±29.155
x = ±29.155 + 40
= 69.155 ---> outside our restriction
or
x = 10.845

To solve this problem, let's break it down step by step:

Step 1: Calculate the area of the plot.
The plot is rectangular and has dimensions of 60 feet by 100 feet. The area of a rectangle is calculated by multiplying the length by the width. So in this case, the area of the plot is 60 feet * 100 feet = 6000 square feet.

Step 2: Calculate the area of the planted garden.
The problem states that the area of the garden is one half of the area of the plot. Since the area of the plot is 6000 square feet, the area of the garden is 6000 square feet / 2 = 3000 square feet.

Step 3: Calculate the dimensions of the garden.
To calculate the dimensions of the garden, we need to find the length and width. Let's assume that the width of the walkway is "x" feet.

The length of the garden with the walkway will be 60 feet - 2x (since there are two walkways on either side).

The width of the garden with the walkway will be 100 feet - 2x (since there are two walkways on top and bottom).

Step 4: Calculate the area of the garden with the walkway.
The area of the garden with the walkway can be calculated by multiplying the length with the width. So it will be (60 feet - 2x) * (100 feet - 2x).

Step 5: Set up the equation.
The problem states that the area of the garden is 3000 square feet. So we can set up the equation:

(60 feet - 2x) * (100 feet - 2x) = 3000 square feet.

Step 6: Solve the equation.
We can now solve the equation. Expanding the equation, we get:

(6000 - 120x - 200x + 4x^2) = 3000.

Simplifying further, we have:

4x^2 - 320x + 3000 = 0.

Step 7: Solve for x.
To solve this quadratic equation, we can factor it or use the quadratic formula. Factoring may not work in this case, so let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a.

For our equation, a = 4, b = -320, and c = 3000. Plugging these values into the quadratic formula, we get:

x = (-(-320) ± √((-320)^2 - 4*4*3000)) / (2*4).

Simplifying further:

x = (320 ± √(102400 - 48000)) / 8.

x = (320 ± √54400) / 8.

x = (320 ± 232.83) / 8.

So we have two possible solutions for x:

x₁ = (320 + 232.83) / 8 ≈ 67.604.

x₂ = (320 - 232.83) / 8 ≈ 9.897.

Since the width of the walkway cannot be negative, we discard x₂ as a solution. Therefore, the width of the walkway is approximately 67.604 feet.