Question

A rectangular garden, 42 feet by 20 feet, is surrounded by a walkway of uniform width. If the total area of the garden and walkway is 1248 square feet, what is the width of the walkway?

Answer 1

The width of the walkway is 3 feet.

Explanation:

Let's call the width of our walkway `x`. We can say that the entire area covered by the walkway and garden is `(20 + 2x)` by `(42 + 2x)`. This can be seen in the diagram below:

Now, we can use the FOIL method to find an expression for the total area of the walkway and path in terms of `x`.

`(2x + 20)(2x + 42)`

`= 2x*2x + 2x*42 + 2x*20 + 20*42`

`= 4x^2 + 84x + 40x + 840`

`= 4x^2 + 124x + 840`

Finally, we know that this area must also be equal to 1248, so we can set these two expression equal and then solve for `x`.

`1248 = 4x^2 + 124x + 840`

`0 = 4x^2 + 124x - 408`

`0 = x^2 + 31x - 102`

This can be factored, since `-102 = (34)(-3)` and `31 = (34) + (-3)`.

`0 = (x - 3)(x + 34)`

By the properties of 0 and multiplication, this gives us these two equations, either of which could be true.

`(x-3) = 0 " " or (x+34) = 0`

`x = 3 " " or " " x = -34`

The width of the path obviously cannot be negative, so the only answer that we are left with is `x=3`.

We can check this by multiplying `26 xx 48` to get `1248`.

Final Answer