Question

Find (a) the arc length and (b) the area of a sector.

Answer 1

a) 23.56 ft (2 dp)

b) 58.90 ft² (2 dp)

Explanation:

Formula

`\textsf{Arc length}=r \theta`

`\textsf{Area of a sector}=(1)/(2)r^2 \theta`

`\quad \textsf{(where r is the radius and}\:\theta\:{\textsf{is the angle in radians)}`

Calculation

Given:

• 
  `\theta=(3 \pi)/(2)`
• r = 5 ft

`\begin{aligned}\implies \textsf{Arc length} & =r \theta\\& = 5\left((3 \pi)/(2)\right)\\& = (15)/(2) \pi \\& = 23.56\: \sf ft\:(2\:dp)\end{aligned}`

`\begin{aligned} \implies \textsf{Area of a sector}& =(1)/(2)r^2 \theta\\\\ & = (1)/(2)(5^2) \left((3 \pi)/(2)\right)\\\\& = (25)/(2)\left((3 \pi)/(2)\right)\\\\ & = (75)/(4) \pi \\\\& = 58.90 \: \sf ft^2\:(2\:dp)\end{aligned}`