IDNLearn.com offers a reliable platform for finding accurate and timely answers. Discover detailed answers to your questions with our extensive database of expert knowledge.
Sagot :
To determine the area of grass that will be watered by the sprinkler, we need to find the area of a sector of a circle. A sector is essentially a slice of the circle, defined by the radius and the central angle.
Here, we are told:
- The radius of the circle is 20 feet.
- The central angle is [tex]\(80^\circ\)[/tex].
First, we need to convert the central angle from degrees to radians because the formula to find the area of a sector typically uses radians. The conversion factor from degrees to radians is:
[tex]\[ 1 \text{ degree} = \frac{\pi \text{ radians}}{180^\circ} \][/tex]
So, converting [tex]\(80^\circ\)[/tex] to radians:
[tex]\[ 80^\circ \times \frac{\pi \text{ radians}}{180^\circ} = \frac{80\pi}{180} = \frac{4\pi}{9} \][/tex]
Now we have the central angle in radians as [tex]\(\frac{4\pi}{9}\)[/tex].
Next, we use the formula for the area of a sector, which is:
[tex]\[ \text{Area of a sector} = \frac{\text{central angle in radians}}{2\pi} \times \pi \times \text{radius}^2 \][/tex]
Substituting the values we have:
[tex]\[ \text{Area} = \frac{\frac{4\pi}{9}}{2\pi} \times \pi \times 20^2 \][/tex]
Simplify inside the formula:
[tex]\[ \text{Area} = \left(\frac{4\pi}{9} \times \frac{1}{2\pi}\right) \times \pi \times 400 \][/tex]
[tex]\[ \text{Area} = \left(\frac{4}{18}\right) \times 400 \][/tex]
[tex]\[ \text{Area} = \left(\frac{2}{9}\right) \times 400 \][/tex]
[tex]\[ \text{Area} = \frac{800}{9} \][/tex]
So, the area of the sector, or the area of grass that will be watered, is:
[tex]\[ \frac{800}{9} \pi \ \text{square feet} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\frac{800}{9} \pi \ \text{ft}^2} \][/tex]
The correct choice is option B.
Here, we are told:
- The radius of the circle is 20 feet.
- The central angle is [tex]\(80^\circ\)[/tex].
First, we need to convert the central angle from degrees to radians because the formula to find the area of a sector typically uses radians. The conversion factor from degrees to radians is:
[tex]\[ 1 \text{ degree} = \frac{\pi \text{ radians}}{180^\circ} \][/tex]
So, converting [tex]\(80^\circ\)[/tex] to radians:
[tex]\[ 80^\circ \times \frac{\pi \text{ radians}}{180^\circ} = \frac{80\pi}{180} = \frac{4\pi}{9} \][/tex]
Now we have the central angle in radians as [tex]\(\frac{4\pi}{9}\)[/tex].
Next, we use the formula for the area of a sector, which is:
[tex]\[ \text{Area of a sector} = \frac{\text{central angle in radians}}{2\pi} \times \pi \times \text{radius}^2 \][/tex]
Substituting the values we have:
[tex]\[ \text{Area} = \frac{\frac{4\pi}{9}}{2\pi} \times \pi \times 20^2 \][/tex]
Simplify inside the formula:
[tex]\[ \text{Area} = \left(\frac{4\pi}{9} \times \frac{1}{2\pi}\right) \times \pi \times 400 \][/tex]
[tex]\[ \text{Area} = \left(\frac{4}{18}\right) \times 400 \][/tex]
[tex]\[ \text{Area} = \left(\frac{2}{9}\right) \times 400 \][/tex]
[tex]\[ \text{Area} = \frac{800}{9} \][/tex]
So, the area of the sector, or the area of grass that will be watered, is:
[tex]\[ \frac{800}{9} \pi \ \text{square feet} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\frac{800}{9} \pi \ \text{ft}^2} \][/tex]
The correct choice is option B.
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Find precise solutions at IDNLearn.com. Thank you for trusting us with your queries, and we hope to see you again.