Calculating the Area of the Arc

Oct 13, 2025 by thesweetdesigns.com 8 views

Understanding the Problem

When it comes to watering grass, a rotating sprinkler head is a common tool used to cover a large area efficiently. However, to ensure that the right amount of water is applied to the grass, it's essential to calculate the area that will be watered. In this problem, we're given a sprinkler head that sprays water as far as 20 feet and is set to cover a central angle of $$80^{\circ}$$. Our goal is to determine the area of grass that will be watered.

Visualizing the Problem

To approach this problem, let's visualize the situation. Imagine a circle with a radius of 20 feet, representing the area that the sprinkler head can cover. The sprinkler head is set to cover a central angle of $$80^{\circ}$$, which means that it will spray water over an arc of this circle. Our task is to find the area of this arc.

Calculating the Area of the Arc

To calculate the area of the arc, we can use the formula for the area of a sector of a circle:

A = \frac{\theta}{360} \pi r^2

where $$A$$ is the area of the sector, $$\theta$$ is the central angle in degrees, $$\pi$$ is a mathematical constant approximately equal to 3.14, and $$r$$ is the radius of the circle.

In this case, we're given that the central angle $$\theta$$ is $$80^{\circ}$$ and the radius $$r$$ is 20 feet. Plugging these values into the formula, we get:

A = \frac{80}{360} \pi (20)^2

Simplifying the Expression

To simplify the expression, we can start by evaluating the fraction:

\frac{80}{360} = \frac{2}{9}

Next, we can square the radius:

(20)^2 = 400

Now, we can substitute these values back into the expression:

A = \frac{2}{9} \pi (400)

Calculating the Final Answer

To calculate the final answer, we can multiply the fraction by the radius squared:

A = \frac{2}{9} \pi (400) = \frac{800}{9} \pi

Since we're asked to provide the answer in terms of $$\pi$$, we can leave the expression as is. However, if we want to provide a numerical value, we can approximate $$\pi$$ as 3.14 and multiply it by the fraction:

A \approx \frac{800}{9} \times 3.14 = 280.89 \, \text{ft}^2

Conclusion

In conclusion, the area of grass that will be watered by the rotating sprinkler head is approximately 280.89 square feet. This calculation assumes that the sprinkler head is set to cover a central angle of $$80^{\circ}$$ and sprays water as far as 20 feet.

Answer Options

Based on our calculation, the correct answer is:

A. $\frac{80}{8} \pi \, \text{ft}^2$ is incorrect, as it does not take into account the central angle of the sprinkler head.
B. $\frac{800}{9} \pi \, \text{ft}^2$ is the correct answer, as it accurately represents the area of grass that will be watered by the sprinkler head.

Final Answer

The final answer is: $\boxed{\frac{800}{9} \pi \, \text{ft}^2}$

Understanding Rotating Sprinkler Heads

Rotating sprinkler heads are a common tool used to water large areas of grass efficiently. However, to ensure that the right amount of water is applied to the grass, it's essential to understand how they work and how to use them effectively.

Q: What is the purpose of a rotating sprinkler head?

A: The primary purpose of a rotating sprinkler head is to water large areas of grass efficiently. They are designed to spray water over a wide area, typically in a circular pattern, to ensure that the grass receives the right amount of water.

Q: How do rotating sprinkler heads work?

A: Rotating sprinkler heads work by using a combination of water pressure and a rotating mechanism to spray water over a wide area. The sprinkler head is typically attached to a hose or a water source, and when the water is turned on, the sprinkler head begins to rotate, spraying water in a circular pattern.

Q: What are the benefits of using a rotating sprinkler head?

A: The benefits of using a rotating sprinkler head include:

Efficient watering: Rotating sprinkler heads can water large areas of grass quickly and efficiently.
Uniform watering: They ensure that the grass receives the right amount of water, reducing the risk of overwatering or underwatering.
Convenience: Rotating sprinkler heads are easy to use and require minimal maintenance.

Q: How do I choose the right rotating sprinkler head for my lawn?

A: To choose the right rotating sprinkler head for your lawn, consider the following factors:

Lawn size: Choose a sprinkler head that is designed for your lawn size.
Water pressure: Make sure the sprinkler head is compatible with your water pressure.
Spray pattern: Consider the type of spray pattern you need, such as a circular or rectangular pattern.

Q: How do I adjust the sprinkler head to cover the right area?

A: To adjust the sprinkler head to cover the right area, follow these steps:

Check the manufacturer's instructions: Consult the manufacturer's instructions for specific guidance on adjusting the sprinkler head.
Adjust the radius: Adjust the radius of the sprinkler head to cover the desired area.
Check the spray pattern: Check the spray pattern to ensure it is even and consistent.

Q: How often should I water my lawn with a rotating sprinkler head?

A: The frequency of watering your lawn with a rotating sprinkler head depends on various factors, including:

Lawn type: Different types of grass require different amounts of water.
Weather conditions: Weather conditions, such as temperature and humidity, can affect the amount of water your lawn needs.
Soil type: The type of soil your lawn is growing in can also affect the amount of water it needs.

Q: Can I use a rotating sprinkler head on slopes or uneven terrain?

A: Yes, you can use a rotating sprinkler head on slopes or uneven terrain. However, you may need to adjust the sprinkler head to ensure it is level and even.

Q: How do I maintain my rotating sprinkler head?

A: To maintain your rotating sprinkler head, follow these steps:

Clean the sprinkler head: Regularly clean the sprinkler head to ensure it is free from debris and mineral buildup.
Check the O-rings: Check the O-rings for wear and tear and replace them as needed.
Lubricate the sprinkler head: Lubricate the sprinkler head to ensure it is running smoothly.

Conclusion

In conclusion, rotating sprinkler heads are a convenient and efficient way to water large areas of grass. By understanding how they work and how to use them effectively, you can ensure that your lawn receives the right amount of water.