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Joe wants to make a path around the outside perimeter of his vegetable garden which is 50’ x 70’. If the path is of uniform width and the entire garden and path are to cover a surface area of 4524 sq feet how wide should the path be

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Facundo3141592idn

We will see that the path should be 4 feet wide.

How wide should the path be?

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:

  • L = 50 ft
  • W = 70 ft

Now if we add a path of width x that surrounds it, the new length and width will be:

  • L' = 50 ft + 2x
  • W' = 70 ft + 2x

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  + 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

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