Assume the width of the path be x.
Length : 20 + 2x,Width : 14 + 2x

The area of the outer rectangle is :
(14 + 2x)(20 + 2x) = 4x² + 68x + 280

The area of the field :
14 × 20 = 280m²

The area of the path is the difference between the two areas :
(4x² + 68x + 280) - 280 = 4x² + 68x

=} 4x² + 68x = 72

=} 4x² + 68x - 72 = 0

Divide through by 4

X² + 17x - 18 = 0

Solving for x :

The roots are : - 1 and 18

=} x ² - x + 18x - 18 = 0

=} x (x - 1) + 18(x - 1) = 0

=} (x - 1)(x + 18) = 0

=} x = 1 or - 18

Hence, as neglected the '-ve' value length of path is 1m.

That's all🙂

Solution:

Let's assume the width of the path to be 'x' meters.

Step 1: Calculate the dimensions of the inner rectangle

To find the dimensions of the inner rectangle, we need to subtract the width of the path from the original length and width of the field.

Length of the inner rectangle = Length of the field - 2 * Width of the path
= 20 - 2x
= 20 - 2x meters

Width of the inner rectangle = Width of the field - 2 * Width of the path
= 14 - 2x
= 14 - 2x meters

Step 2: Calculate the area of the inner rectangle

The area of the inner rectangle can be calculated by multiplying its length and width.

Area of the inner rectangle = (20 - 2x) * (14 - 2x)
= 280 - 40x - 28x + 4x^2
= 4x^2 - 68x + 280

Step 3: Calculate the area of the path

The area of the path is given as 111 sq m.

Area of the path = Area of the field - Area of the inner rectangle
= (20 * 14) - (4x^2 - 68x + 280)
= 280 - (4x^2 - 68x + 280)
= 280 - 4x^2 + 68x - 280
= -4x^2 + 68x

Step 4: Set up the quadratic equation

The area of the path can be expressed as a quadratic equation in terms of 'x'.

-4x^2 + 68x = 111

Rearranging the equation, we get:

-4x^2 + 68x - 111 = 0

Step 5: Solve the quadratic equation

To solve the quadratic equation, we can use factoring, completing the square, or the quadratic formula.

In this case, let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Here, a = -4, b = 68, and c = -111.

x = (-68 ± √(68^2 - 4*(-4)*(-111))) / (2*(-4))
x = (-68 ± √(4624 - 1776)) / (-8)
x = (-68 ± √2848) / (-8)
x = (-68 ± 53.37) / (-8)

So, the possible values of 'x' are:

x = (-68 + 53.37) / (-8)
x = (-14.63) / (-8)
x ≈ 1.83

or

x = (-68 - 53.37) / (-8)
x = (-121.37) / (-8)
x ≈ 15.17

Step 6: Determine the width of the path

Since the width cannot be negative, we discard the negative value of 'x'.

So, the width of the path is approximately 1.83 meters.