Question

find the area of each of the following shaded regions. assume all arcs are circular with centers marked.
(simplify your answer. do not factor. type an exact answer in terms of π.)
b. the area is (2π + 4) cm²
(simplify your answer. do not factor. type an exact answer in terms of π.)
c. the area is (2π) cm²
(simplify your answer. do not factor. type an exact answer in terms of π.)
d. the area is
(simplify your answer. do not factor. type an exact answer in terms of π.)

Explanation:

Step1: Analyze the figure

The shaded region in part d can be found by calculating the area of 8 semicircles (which is equivalent to 4 full - circles) and subtracting the area of the square. The radius of each semicircle \(r=\frac{30}{2}=15\) cm, and the side of the square \(s = 30\) cm.

Step2: Calculate the area of the 4 circles

The area of a circle is \(A=\pi r^{2}\). For \(r = 15\) cm, the area of 4 circles is \(A_{circles}=4\times\pi\times(15)^{2}=4\times\pi\times225 = 900\pi\)

Step3: Calculate the area of the square

The area of a square is \(A_{square}=s^{2}\). For \(s = 30\) cm, \(A_{square}=30\times30=900\)

Step4: Calculate the area of the shaded region

The area of the shaded region \(A=A_{circles}-A_{square}\)

Answer:

\(900\pi - 900\)