Question

margaret needs to put a new gutter on one side of her roof. the shape of her roof is made up of two right triangles that are on each side of a square. if the area of the square is 8100 ft² and the base of each of the triangles is 60 ft, what is the total length of the gutter shell need to replace? answer

Explanation:

Step1: Find the side length of the square

The area formula of a square is \(A = s^{2}\), where \(A\) is the area and \(s\) is the side - length. Given \(A=8100\space ft^{2}\), then \(s=\sqrt{8100}\).
Since \(\sqrt{8100}=\sqrt{90\times90}\), so \(s = 90\space ft\).

Step2: Use the Pythagorean theorem for the right - triangle

For a right - triangle, by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 60\space ft\), \(b = 90\space ft\) (the side of the square is the height of the right - triangle in this context), and \(c\) is the hypotenuse of the right - triangle.
\(c=\sqrt{60^{2}+90^{2}}=\sqrt{3600 + 8100}=\sqrt{11700}=\sqrt{900\times13}=30\sqrt{13}\space ft\)

Step3: Calculate the total length of the gutter

The total length of the gutter \(L=2c + s\).
Substitute \(c = 30\sqrt{13}\space ft\) and \(s = 90\space ft\) into the formula:
\(L=2\times30\sqrt{13}+90=60\sqrt{13}+90\approx60\times3.606+90=216.36+90=306.36\space ft\)

Answer:

\(306.36\space ft\)