Algebra Solutions
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An open rectangular gutter is to be made by turning up the sides of a piece of metal 18 inches wide. If the cross-section area of the resulting gutter is 36 square inches, find the gutter's depth.
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 \(6 ~inches\)