Question

1. a = 1/2(7)(24) a = 1/2h(b1 + b2) 8 7 16 7 8 168 - 16π a = p =

Explanation:

Step1: Analyze the figure for area

The figure can be seen as two trapezoids with bases \(b_1 = 8\), \(b_2=16\) and height \(h = 7\), and two semi - circles (which together form a full circle) with diameter \(d = 8\) (radius \(r = 4\)). The area of a trapezoid is \(A_{t}=\frac{1}{2}(b_1 + b_2)h\) and the area of a circle is \(A_{c}=\pi r^{2}\). The total area of the two trapezoids is \(2\times\frac{1}{2}(8 + 16)\times7=168\), and the area of the circle is \(\pi\times4^{2}=16\pi\). So the area of the shaded region \(A = 168-16\pi\).

Step2: Analyze the figure for perimeter

The perimeter consists of the non - parallel sides of the trapezoids and the circumferences of the semi - circles. The non - parallel sides of each trapezoid can be found using the Pythagorean theorem. But we can also note that the perimeter contribution from the trapezoids is the sum of the non - parallel sides. The length of the non - parallel side of each trapezoid is \(\sqrt{7^{2}+4^{2}}=\sqrt{49 + 16}=\sqrt{65}\). The sum of the non - parallel sides of the two trapezoids is \(4\sqrt{65}\), and the circumference of the two semi - circles (a full circle) is \(C=\pi d=8\pi\). So the perimeter \(P=4\sqrt{65}+8\pi\).

Answer:

\(A = 168-16\pi\), \(P=4\sqrt{65}+8\pi\)