Question

30. Landscaping Calculate the area (in square feet) of a flower garden shaped like a circular sector withradius 60 ft and central angle 33 degrees.31. In problem 30; if shrubs are planted every 2 ft along the outer border of the garden, how many shrubsare needed?

Answer 1

`\begin{gathered} 30)\text{Area}_(sc)=1036.72ft^2 \\ 31)Number\text{ of shrubs=}78 \end{gathered}`

Step-by-step explanation

the area of a circular sector is given by

`\begin{gathered} \text{Area}_(sc)=(\theta)/(360)\pi r^2 \\ \text{where r is the radius and }\theta\text{ is the angle in degr}ees \end{gathered}`

then

Step 1

Let

`\begin{gathered} \text{radius}=\text{ 60 ft} \\ \text{angle}=33\text{ \degree} \end{gathered}`

now, replace in the formula

`\begin{gathered} \text{Area}_(sc)=(\theta)/(360)\pi r^2 \\ \text{Area}_(sc)=(33)/(360)\pi(60ft)^2 \\ \text{Area}_(sc)=1036.72ft^2 \\ \text{rounded} \\ \text{Area}_(sc)=1036.72ft^2 \end{gathered}`

Step 2

if shrubs are planted every 2 ft along the outer border of the garden, how many shrubs

are needed?

to figure out this, we need to take the perimeter of the circular sector and divide by 2 ft, to get the total number of shrubs in the border

so,

`\text{perimeter}=(2\cdot\text{radius)}+length\text{ of arc}`

so, we need to find the length of the arc

the length of the arc is given by

`l=(2\pi r)/(360)\cdot\theta`

replace.

`\begin{gathered} l=(2\pi r)/(360)\cdot\theta \\ l=(2\pi\cdot60)/(360)\cdot33 \\ l=34.55 \end{gathered}`

finally, replace in the perimeter formula

`\begin{gathered} \text{perimeter}=(2\cdot\text{radius)}+length\text{ of arc} \\ \text{perimeter}=(2\cdot60ft\text{)}+34.55 \\ \text{perimeter}=120\text{ ft+34.55 ft} \\ \text{Perimeter}=154.55\text{ ft} \end{gathered}`

divde by 2 to know the numbers of shrubs

`\begin{gathered} Numbver\text{ of shrubs=}\frac{\text{ perimeter}}{2})=\frac{154.55\text{ ft}}{2} \\ Numbver\text{ of shrubs=}77.275\text{ ft} \\ Numbver\text{ of shrubs=}78 \end{gathered}`