a gravel path of equal width is to be built aroung a 4' by 4' square garden. How wide can the path be if there is enough gravel for 180 ft^2

Question

whpalmer4 · Answer

First, start with a diagram.|dw:1372617569724:dw|

whpalmer4 · Answer

the area to be filled with gravel is the region surrounding the 4x4 garden.  The area of the garden + path is simply $x^2$, and the area of the garden is $4^2$ (because the area of a rectangle = $\ell * w$).

That means the area filled with gravel is the difference between the overall area ($x^2$) and the garden area ($4^2$), and we know that area is 180 sq ft.

$$x^2-4^2=180$$Solve the equation for $x$.  There are two solutions, but one of them is nonsensical for this problem.

Now when you have found the value of $x$ remember that that is the entire width of the garden plus the path on each side, NOT the width of the path.  You'll need to find the width of the path by subtracting the width of the garden and dividing the remaining amount in half.

It is also possible to write the equations to have $x$ represent the width of the path directly.  That makes for a slightly more difficult equation to solve, but then you don't have to postprocess the solution.