Question

find the area of the figure. round to the nearest hundredths when necessary. 16.9 ft 11.9 ft answer attempt 1 out of 2

Explanation:

Step1: Find the height of the right - triangle (also the radius of the semicircle)

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). Let the height \(h\), \(c = 16.9\), \(a=11.9\). Then \(h=\sqrt{16.9^{2}-11.9^{2}}=\sqrt{(16.9 + 11.9)(16.9 - 11.9)}=\sqrt{28.8\times5}=\sqrt{144}=12\) ft.

Step2: Calculate the area of the right - triangle

The area formula for a triangle is \(A_{\triangle}=\frac{1}{2}\times base\times height\). Here, base \(b = 11.9\) ft and height \(h = 12\) ft. So \(A_{\triangle}=\frac{1}{2}\times11.9\times12=71.4\) \(ft^{2}\).

Step3: Calculate the area of the semicircle

The area formula for a circle is \(A=\pi r^{2}\), for a semicircle \(A_{semicircle}=\frac{1}{2}\pi r^{2}\). Since \(r = 12\) ft, \(A_{semicircle}=\frac{1}{2}\times\pi\times12^{2}=\frac{1}{2}\times\pi\times144 = 72\pi\approx72\times3.14 = 226.08\) \(ft^{2}\).

Step4: Calculate the total area of the figure

\(A = A_{\triangle}+A_{semicircle}\). So \(A=71.4 + 226.08=297.48\) \(ft^{2}\).

Answer:

\(297.48\) \(ft^{2}\)