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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{80}{9} \pi \, \text{ft}^2[/tex]
B. [tex] \frac{200}{9} \pi \, \text{ft}^2[/tex]
C. [tex] \frac{800}{9} \pi \, \text{ft}^2[/tex]
D. [tex] \frac{760}{9} \pi \, \text{ft}^2[/tex]



Answer :

GinnyAnswer
To solve this problem, we need to determine the area of grass watered by a rotating sprinkler head. Here's a step-by-step solution:

1. Identify the radius of the circle:
The radius is given as 20 feet.

2. Determine the central angle in degrees:
The central angle covered by the sprinkler is [tex]\( 80^\circ \)[/tex].

3. Find the area of the entire circle:
The formula for the area of a circle is [tex]\( A = \pi r^2 \)[/tex], where [tex]\( r \)[/tex] is the radius.
[tex]\[ A_{\text{full}} = \pi \times (20)^2 = \pi \times 400 = 400\pi \text{ square feet} \][/tex]

4. Calculate the fraction of the circle being covered by the sprinkler:
The fraction of the circle covered by the [tex]\( 80^\circ \)[/tex] spray is given by the ratio of the central angle to the full circle (which is [tex]\( 360^\circ \)[/tex]).
[tex]\[ \text{Fraction of circle} = \frac{80}{360} = \frac{2}{9} \][/tex]

5. Determine the area being watered:
Multiply the fraction of the circle by the total area of the circle to obtain the area being watered.
[tex]\[ A_{\text{watered}} = \frac{2}{9} \times 400\pi = \frac{800\pi}{9} \text{ square feet} \][/tex]

Therefore, the area of grass that will be watered is [tex]\( \frac{800}{9} \pi \)[/tex] square feet.

The correct answer is:
[tex]\[ \boxed{\frac{800}{9} \pi \text{ ft}^2} \][/tex]