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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][tex]$80^{\circ}$[/tex][/tex]. What area of grass will be watered?

A. [tex]\frac{\pi}{2} ft^2[/tex]
B. [tex]\frac{80}{360} \pi \cdot 20^2 ft^2[/tex]
C. [tex]\frac{1}{4} \pi \cdot 20^2 ft^2[/tex]
D. [tex]\frac{20}{2} \pi ft^2[/tex]

Sagot :

To determine the area of grass watered by the rotating sprinkler head, we need to calculate the area of a sector of a circle. The formula for the area [tex]\( A \)[/tex] of a sector with a central angle [tex]\( \theta \)[/tex] (in radians) and radius [tex]\( r \)[/tex] is given by:

[tex]\[ A = \frac{1}{2} r^2 \theta \][/tex]

We are given:
- The radius [tex]\( r \)[/tex] of the circle is 20 feet.
- The central angle [tex]\( \theta \)[/tex] is [tex]\( 80^\circ \)[/tex].

Next, we need to convert the angle from degrees to radians because the formula requires the angle in radians. The conversion from degrees to radians is given by:

[tex]\[ \theta(\text{radians}) = \theta(\text{degrees}) \times \frac{\pi}{180} \][/tex]

So, converting [tex]\( 80^\circ \)[/tex] to radians:

[tex]\[ \theta(\text{radians}) = 80 \times \frac{\pi}{180} = \frac{80\pi}{180} = \frac{4\pi}{9} \][/tex]

Now, plugging these values into the area formula:

[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]

Using numerical computation (previously determined):

- The central angle in radians is approximately [tex]\( 1.396 \)[/tex] radians.
- The area watered is approximately [tex]\( 279.253 \)[/tex] square feet.

Thus, the correct area of grass that will be watered by the sprinkler is approximately [tex]\( 279.2527 \, \text{square feet} \)[/tex], corresponding to the numerical result previously determined. None of the provided options directly correspond to this specific computed area, but the step-by-step rationale, involves verifying calculations and considering any potential oversight in the provided question options.
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