Question

find the area of the shaded region.
x + 8
x + 3

Explanation:

Step1: Calculate area of rectangle

The formula for the area of a rectangle is \( A_{\text{rectangle}} = \text{length} \times \text{width} \). Here, length is \( x + 8 \) and width is \( x + 3 \). So, \( A_{\text{rectangle}}=(x + 8)(x + 3) \). Expanding this, we get \( x^{2}+3x + 8x+24=x^{2}+11x + 24 \).

Step2: Calculate area of circle

The formula for the area of a circle is \( A_{\text{circle}}=\pi r^{2} \), where \( r = x \). So, \( A_{\text{circle}}=\pi x^{2} \).

Step3: Calculate area of shaded region

The shaded region's area is the area of the rectangle minus the area of the circle. So, \( A_{\text{shaded}}=A_{\text{rectangle}}-A_{\text{circle}}=(x^{2}+11x + 24)-\pi x^{2}=(1 - \pi)x^{2}+11x + 24 \).

Answer:

\((1 - \pi)x^{2}+11x + 24\)