Perimeter of the gutter \(P = 2d +w = 18\)

Area of the gutter: \(A = dw = 36\)

\(w = \frac{36}{d}\)

\(P = 2d + \frac{36}{d} = 18\)

\(18d =2d^2 +36 \)

\(d^2 - 9d + 18 =0 \)

\(d_1 = \frac{9 +\sqrt{9^2-4(18)}}{2}\)

\(d_1 = 6 ~inches\)

\(d_2 = \frac{9 -\sqrt{9^2-4(18)}}{2}\)

\(d_2 =3 ~inches\)

Check what value is correct (answer).

\(w = \frac{36}{6} = 6\)

\(w = \frac{36}{3} = 12\)

\(A = 12x3 = 6x6 = 36\) and \(P = 2(3) + 12 = 2(6) +6 = 18\)

Since both 6" and 3" gives a correct area and perimeter the answer is either \(3 ~inches\) or \(6 ~inches\)