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A rotating sprinkler head sprays water as far as 20 feet. The head is set to cover a central angle of [tex]$80^{\circ}$[/tex]. What area of grass will be watered?

A. [tex] \frac{760}{9} \pi \, \text{ft}^2 [/tex]

B. [tex] \frac{80}{9} \pi \, \text{ft}^2 [/tex]

C. [tex] \frac{200}{9} \pi \, \text{ft}^2 [/tex]

D. [tex] \frac{800}{9} \pi \, \text{ft}^2 [/tex]

Sagot :

Let's solve the problem step-by-step to determine the area of grass that will be watered by the rotating sprinkler head.

Step 1: Understand the Problem
- The sprinkler sprays water up to a radius of 20 feet.
- It covers a central angle of [tex]\(80^\circ\)[/tex].

Step 2: Convert the Angle from Degrees to Radians
To use the formula for the area of a sector, we need to convert the angle from degrees to radians. The conversion factor is:
[tex]\[ 1 \text{ degree} = \frac{\pi}{180} \text{ radians} \][/tex]

So, for [tex]\(80^\circ\)[/tex]:
[tex]\[ 80^\circ \times \frac{\pi}{180} = \frac{80\pi}{180} = \frac{4\pi}{9} \text{ radians} \][/tex]

Step 3: Use the Formula for the Area of a Sector
The formula for the area [tex]\(A\)[/tex] of a sector with radius [tex]\(r\)[/tex] and angle [tex]\(\theta\)[/tex] in radians is:
[tex]\[ A = \frac{1}{2} r^2 \theta \][/tex]

Here, [tex]\(r = 20 \text{ feet}\)[/tex] and [tex]\(\theta = \frac{4\pi}{9} \text{ radians} \)[/tex].

Step 4: Plug in the Values
[tex]\[ A = \frac{1}{2} \times (20)^2 \times \frac{4\pi}{9} \][/tex]
[tex]\[ A = \frac{1}{2} \times 400 \times \frac{4\pi}{9} \][/tex]
[tex]\[ A = 200 \times \frac{4\pi}{9} \][/tex]
[tex]\[ A = \frac{800 \pi}{9} \][/tex]

So, the area of the grass that will be watered is:
[tex]\[ \boxed{\frac{800\pi}{9} \text{ square feet}} \][/tex]

Therefore, the correct answer is [tex]\(D. \frac{800}{9} \pi \text{ ft}^2\)[/tex].