A rectangular garden measures 13 meters by 17 meters and has a cement walkway around its perimeter, as shown. The width of the walkway remains constant on all four sides. The garden and walkway have a combined area of 396 square meters. Enter an equation that could be used to help determine the width, w, of the walkway. Enter your answer in the response box. Also determine the width, in meters, of the walkway.

The equation is

W^2 + 30W - 175 = 0

And the width of the walkway = 5 meters

Step-by-step explanation: For the square garden we have the dimensions as 17 meters by 13 meters. Also there is a walkway that borders all four sides of the rectangular garden. So the dimensions of the garden plus walkway, can be derived as;

L= 17 + W and

W = 13 + W

If the area of the entire shape (garden plus walkway inclusive) is given as 396 square meters, then we can write the equation as follows

Area of a rectangle = L x W

Area of a rectangle= (17 + W) x (13 + W)

396 = 221 + 17W + 13W + W^2

396 - 221 = 30W + W^2

(Remember that when a positive value crosses to the other side of an equation it becomes negative and vice versa)

175 = 30W + W^2

W^2 + 30W - 175 = 0

Having found the equation (a quadratic equation) that can be used to determine the width of the walkway, which is W, we can now proceed to calculate W as follows

W^2 + 30W - 175 = 0

By factorizing we now arrive at

W^2 +35W - 5W - 175 = 0

(W + 35) (W - 5) = 0

That means,

Either W + 35 = 0, hence W = -35

OR

W - 5 = 0, hence W = 5

Since W cannot be a negative value, we know that W= 5

Therefore the width of the walkway is 5 meters

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