We will see that the path should be 4 feet wide.
For a rectangle of length L and width W, the area is:
A = L*W
In this case, we know that the dimensions of the garden are:
Now if we add a path of width x that surrounds it, the new length and width will be:
And the area of this rectangle must be 4524 ft^2, then we have:
(50 ft + 2x)*(70 ft + 2x) = 4524 ft^2
Now we can solve this for x.
Ignoring the units, so the math is easier to read, we have:
(50 + 2x)*(70 Â + 2x) = 4524
50*70 + 50*2x + 2x*70 + 2x*2x = 4524
3500 + 240x + 4x^2 = 4524
4x^2 + 240x + 3500 - 4524 = 0
4x^2 + 240x - 1024 = 0
Now to simplify the quadratic equation, we can divide both sides by 4 to get:
x^2 + 60x  + 256 = 0
The solutions of the quadratic equation are given by Bhaskara's formula:
[tex]x = \frac{-60 \pm \sqrt{60^2 - 4*1*(-256)} }{2*1} \\\\x = \frac{-60 \pm 68 }{2}[/tex]
We only care for the positive solution (the negative one does not make physical sense).
x = (-60 + 68)/2 = 4
So the width of the path must be 4 ft.
If you want to learn more about quadratic equations, you can read:
https://brainly.com/question/1214333
