Question

Find the area of the segment. also, what is the arc length of AB (curved line over AB)

Answer 1

The area of the shaded region can be obtained by subtraction of the triangle ΔABC from the circular region CAB.

First, for the triangle:

We need to find the length of CH and HB (the height and half of the base of the triangle) in order to calculate the area of the triangle. Using trigonometric functions:

`\begin{gathered} \sin50\degree=(HB)/(18)\Rightarrow HB=18\sin50\degree \\ \\ \cos50\degree=(CH)/(18)\Rightarrow CH=18\cos50\degree \end{gathered}`

The area of the triangle is given by:

`A_(triangle)=(base\cdot height)/(2)=18\cos50\degree\cdot18\sin50\degree=324\sin50\degree\cos50\degree`

We know that sin(2x) = 2*sin(x)*cos(x). Then:

`A_(triangle)=162\sin100\degree`

Now, for the circular region, we have the formula:

`A_(circ)=(\theta\cdot r^2)/(2)`

Where θ is the central angle and r is the radius. From the problem, we identify:

`\begin{gathered} \theta=100\degree=100\degree\cdot(\pi)/(180\degree)=(5\pi)/(9) \\ \\ r=18\text{ ft} \end{gathered}`

Using the formula:

`A_(circ)=(5\pi/9\cdot18^2)/(2)=90\pi`

Finally, we subtract A_triangle from A_circ:

`\begin{gathered} A_(shaded)=90\pi-162\sin100\degree \\ \\ \therefore A_(shaded)\approx123.204\text{ ft}{}^2 \end{gathered}`

For the arc length AB, we use the formula:

`AB=\theta\cdot r`

Using the values for the angle and the radius:

`\begin{gathered} AB=(5\pi)/(9)\cdot18 \\ \\ \therefore AB=10\pi\approx31.416\text{ ft} \end{gathered}`

Answer:

`\begin{gathered} \text{ Area }=123.204\text{ ft}^2 \\ \\ \text{ Arc length }=31.416\text{ ft} \end{gathered}`