For this problem, please see the attached picture to illustrate the problem. The garden is represented by the inner rectangle having dimensions of 16×12 feet. A path is added that surrounds the garden. The path creates a uniform margin around the garden with equal spacing denoted as x feet. To determine x, we use the total area of the outer rectangle which is equal to 320 square feet.
We know that the area of a rectangle is equal to length times width. The length to be used here is 16 feet subtracted with two x feet of path accounting for the two corners. The same applies for the width. Therefore, the equation for the area is
A = (16 - 2x)(12 - 2x)
Finally, we equate area to 320 to solve for x:
320 = (16 - 2x)(12 - 2x)
320 = 192 - 124x - 32x + 4x²
320 = 4x² - 56x + 192
Dividing the whole equation by 4,
x² - 14x + 48 = 80
x² - 14x + 48 - 80 = 0
x² - 14x - 32 = 0
Factoring the polynomial,
(x-16)(x+2) = 0
x = 16, -2
From the two roots, we choose the positive root as our solution. Therefore, the path should be 16 ft.
