Question

You are planting a vegetable garden on a plot of land that is a sector of a circle, as shown below.

You want fencing along only the curved edge of the garden.
a. Use the figure to find the length of fencing you will need.
15 ft
75°
b. How much area will be available for planting?

Answer 1

(a) 19.63 ft

(b) 147.26 ft^2

Explanation:

(a) 75/360 degrees . 2 . pi . 15 = 19.63

(b) 75/360 degrees. pi . 15^2 = 147.26

Answer 2

a) 19.63 ft (2 dp)

b) 147.26 ft² (2 dp)

Explanation:

To find the length of the curved fence, use the formula for arc length of a circle.

To find the area of the vegetable garden, use the formula for area of a sector of a circle.

Formula

`\textsf{Arc length}=2 \pi r\left((\theta)/(360^(\circ))\right)`

`\textsf{Area of a sector}=\left((\theta)/(360^(\circ))\right) \pi r^2`

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

Calculation

Given:

• 
  `\theta` = 75°
• r = 15 ft

`\begin{aligned}\implies \textsf{Arc length} &=2 \pi (15)\left((75^(\circ))/(360^(\circ))\right)\\ & = 30 \pi \left((5)/(24)\right)\\ & = (25)/(4) \pi \\ & = 19.63\: \sf ft\:(2\:dp)\end{aligned}`

`\begin{aligned} \implies \textsf{Area of a sector}& =\left((75^(\circ))/(360^(\circ))\right) \pi (15)^2\\& = \left((5)/(24)\right)\pi \cdot 225\\& = (375)/(8) \pi\\& = 147.26\: \sf ft^2 \:(2\:dp)\end{aligned}`