ANSWER:
b. 32.70531 square inches
STEP-BY-STEP EXPLANATION:
We know the length of the side of polygon A, so we calculate the length of the other sides, knowing their relation to the perimeter.
Polygon C = 1 inch
Polygon A = 4 * 1 inches = 4 inches
Polygon B = 1/2 * 4 inches = 2 inches
The area of the heptagon is calculated using the formula:
[tex]A=\frac{7\cdot a^2}{4\cdot\:\tan\left(\frac{180°}{7}\right)}[/tex]
We obtain this area from the following:
By means of trigonometric ratios we obtain the formula.
We use the formula to calculate the Area of each polygon, like this:
[tex]\begin{gathered} A_A=\frac{7\cdot4^2}{4\cdot\:\tan\left(\frac{180°}{7}\right)}=58.14259 \\ \\ \\ A_B=\frac{7\cdot\:2^2}{4\cdot\:\:\tan\left(\frac{180°}{7}\right)}=14.53564 \\ \\ \\ A_C=\frac{7\cdot\:1^2}{4\cdot\:\:\tan\left(\frac{180°}{7}\right)}=3.63391 \end{gathered}[/tex]
We calculate the differences between the areas of A and B and then the difference between the areas of B and C, and finally the difference between the areas obtained
[tex]\begin{gathered} A_1=A_A-A_B=_58.14259-14.53564=43.60695 \\ \\ A_2=A_B-A_C=14.53564-3.63391=10.90173 \\ \\ A=A_1-A_2=43.60695-10.90173=32.70522\cong32.70531 \end{gathered}[/tex]
Therefore, the correct answer is b. 32.70531 square inches
