Question

HELP AS SOON AS POSSIBLE PLEASE.

Answer 1

Answer for 13. :

`(1)/(3)` ft

Answer for 14. :

`162√(2)` - 81 cm^2

Answer 2

13) 8.7 ft²

14) 114.6 cm²

Explanation:

Question 13

The area of the shaded region can be calculated by subtracting the area of the hexagon from the area of the circle.

The formulas for the area of a circle and the area of a regular hexagon are:

`\boxed{\begin{minipage}{3.9 cm}\underline{Area of a circle}\\\\$\vphantom{(3√(3))/(2) }A=\pi r^2$\\\\where $r$ is the radius.\\\end{minipage}}`
`\boxed{\begin{minipage}{4.1 cm}\underline{Area of a regular hexagon}\\\\$A=(3√(3))/(2) r^2$\\\\where $r$ is the radius.\\\end{minipage}}`

The circle and hexagon both have a radius of 4 ft.

Therefore:

`\begin{aligned}\textsf{Shaded area}&=\pi r^2 - (3√(3))/(2)r^2\\\\&=\pi \cdot 4^2 - (3√(3))/(2)\cdot 4^2\\\\&=16\pi - (3√(3))/(2) \cdot 16\\\\&=16\pi - (48√(3))/(2) \\\\&=16\pi - 24√(3)\\\\&=8.69626307...\\\\&=8.7\; \sf ft^2\end{aligned}`

Therefore, the area of the shaded region is 8.7 ft² (nearest tenth).

`\hrulefill`

Question 14

The shaded region is made up of 4 congruent isosceles triangles.

The apex angle of each triangle is the interior angle of a regular octagon, 135°, and the congruent sides measure 9 cm.

The formula for an isosceles triangle is

`\boxed{\begin{minipage}{8 cm}\underline{Area of an isosceles triangle}\\\\$A=(1)/(2)s^2 \sin \theta$\\\\where:\\ \phantom{w} $\bullet$ $s$ is the congruent side length.\\\phantom{w} $\bullet$ $\theta$ is the angle between the congruent sides.\\\end{minipage}}`

Therefore, the area of the shaded region is:

`\begin{aligned}\textsf{Shaded area}&=4 \cdot (1)/(2) \cdot 9^2 \cdot \sin 135^(\circ)\\\\&=4 \cdot (1)/(2) \cdot 81 \cdot (√(2))/(2)\\\\&=2\cdot 81 \cdot (√(2))/(2)\\\\&=162 \cdot (√(2))/(2)\\\\&=81√(2)\\\\&=114.6\; \sf cm^2\;(2\;d.p.)\end{aligned}`

Therefore, the area of the shaded region is 114.6 cm² (nearest tenth).