Question

what is the area of the shaded region (in brown)? multiple - choice 2 points 67.5 cm² 64.9 cm² 74.3 cm² 78.0 cm²

Explanation:

Step1: Calculate area of rectangle

The area formula for a rectangle is $A = l\times w$. Here, $l = 12$ in and $w = 9$ in, so $A_{rectangle}=12\times9 = 108$ in².

Step2: Calculate area of semi - circle

The area formula for a full - circle is $A=\pi r^{2}$, and for a semi - circle it's $A_{semicircle}=\frac{1}{2}\pi r^{2}$. Given $r = \frac{8}{2}=4$ in, $A_{semicircle}=\frac{1}{2}\pi\times4^{2}=\frac{1}{2}\pi\times16 = 8\pi\approx8\times3.14 = 25.12$ in².

Step3: Calculate area of triangle

The area formula for a triangle is $A=\frac{1}{2}bh$. Here, $b = 4$ in and $h = 4$ in, so $A_{triangle}=\frac{1}{2}\times4\times4=8$ in².

Step4: Calculate area of shaded region

$A_{shaded}=A_{rectangle}-A_{semicircle}-A_{triangle}=108 - 25.12-8=74.88\approx74.9$ in².

Answer:

$64.9$ in² is incorrect. The closest correct value among the options (assuming there was a rounding difference in the options) is $74.3$ in². But if we recalculate precisely, the area of the shaded region is approximately $74.88$ in². If we assume the options have some approximation differences, the best - fitting option from the given ones would be $74.3$ in².