Question

find the area of the shaded region. (assume that given measurements are exact, and round approximate answers to the nearest tenth of a square unit. assume ( a = 2 mathrm{~cm} ) and ( b = 3 mathrm{~cm} ).)

Explanation:

Step1: Calculate the area of the rectangle

The formula for the area of a rectangle is \(A_{rectangle}=a\times b\). Given \(a = 2\) cm and \(b=3\) cm, then \(A_{rectangle}=2\times3=6\) \(cm^{2}\).

Step2: Calculate the area of the semicircle

The diameter of the semicircle is equal to \(a = 2\) cm, so the radius \(r=\frac{a}{2}=\frac{2}{2} = 1\) cm. The formula for the area of a full - circle is \(A_{circle}=\pi r^{2}\), and for a semicircle \(A_{semicircle}=\frac{1}{2}\pi r^{2}\). Substituting \(r = 1\) cm, we get \(A_{semicircle}=\frac{1}{2}\times\pi\times(1)^{2}=\frac{\pi}{2}\approx\frac{3.14}{2}=1.57\) \(cm^{2}\).

Step3: Calculate the total area of the shaded region

The total area \(A = A_{rectangle}+A_{semicircle}\). Substituting the values from Step 1 and Step 2, \(A=6 + 1.57=7.57\approx7.6\) \(cm^{2}\).

Answer:

\(7.6\) \(cm^{2}\)