Calculating the Area of Grass Watered

Oct 13, 2025 by miodeya.com 55 views

Understanding the Problem

When a rotating sprinkler head sprays water, it covers a specific area of grass. The problem requires us to find the area of grass that will be watered by the sprinkler head. To solve this problem, we need to use the concept of circular sectors and the formula for the area of a circle.

Given Information

The sprinkler head sprays water as far as 20 feet.
The head is set to cover a central angle of $80^{\circ}$ .

Calculating the Area of Grass Watered

To find the area of grass watered, we need to use the formula for the area of a circular sector:

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

where $A$ is the area of the sector, $\theta$ is the central angle in degrees, and $r$ is the radius of the circle.

In this problem, we are 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 expression inside the parentheses:

(20)^2 = 400

Now, we can substitute this value back into the expression:

A = \frac{80}{360} \pi (400)

Evaluating the Expression

To evaluate the expression, we can start by simplifying the fraction:

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

Now, we can substitute this value back into the expression:

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

Multiplying the Terms

To multiply the terms, we can start by multiplying the fraction by the value inside the parentheses:

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

Evaluating the Product

To evaluate the product, we can start by multiplying the fraction by the value:

\frac{2}{9} \cdot 400 = \frac{800}{9}

Now, we can multiply this value by $\pi$ :

\frac{800}{9} \pi

Conclusion

The area of grass watered by the sprinkler head is $\frac{800}{9} \pi \, \text{ft}^2$ . This is the correct answer.

Comparison with Other Options

Let's compare our answer with the other options:

A. $\frac{760}{9} \pi \, \text{ft}^2$ B. $\frac{800}{9} \pi \, \text{ft}^2$ C. $\frac{820}{9} \pi \, \text{ft}^2$

Our answer, $\frac{800}{9} \pi \, \text{ft}^2$ , matches option B.

Final Answer

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

Understanding Rotating Sprinkler Heads

Rotating sprinkler heads are a common feature in many residential and commercial irrigation systems. They are designed to spray water over a wide area, providing an efficient and effective way to water lawns, gardens, and other outdoor spaces.

Q&A

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

A: The primary purpose of a rotating sprinkler head is to spray water over a wide area, providing an efficient and effective way to water lawns, gardens, and other outdoor spaces.

Q: How does a rotating sprinkler head work?

A: A rotating sprinkler head works by using a series of gears and a motor to rotate the sprinkler head in a circular motion. As the sprinkler head rotates, it sprays water over a wide area, providing an even distribution of water.

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 provide an efficient way to water lawns, gardens, and other outdoor spaces.
Even distribution: The rotating motion of the sprinkler head ensures an even distribution of water, reducing the risk of overwatering or underwatering.
Reduced water waste: Rotating sprinkler heads help to reduce water waste by providing a precise and controlled distribution of water.

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

A: When choosing a rotating sprinkler head, consider the following factors:

Area to be watered: Choose a sprinkler head that is designed to water the area you need to cover.
Water pressure: Choose a sprinkler head that is designed to work with your water pressure.
Flow rate: Choose a sprinkler head that provides the right flow rate for your needs.

Q: How do I install a rotating sprinkler head?

A: Installing a rotating sprinkler head is a relatively simple process that requires the following steps:

Choose a location: Choose a location for the sprinkler head that is level and secure.
Connect the water supply: Connect the water supply to the sprinkler head.
Adjust the sprinkler head: Adjust the sprinkler head to ensure it is properly aligned and set to the correct angle.

Q: How do I maintain a rotating sprinkler head?

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

Regularly inspect the sprinkler head: Regularly inspect the sprinkler head to ensure it is working properly.
Clean the sprinkler head: Clean the sprinkler head regularly to ensure it is free from debris and mineral buildup.
Replace worn parts: Replace worn parts, such as the sprinkler head or gears, as needed.

Q: What are some common problems with rotating sprinkler heads?

A: Some common problems with rotating sprinkler heads include:

Leaks: Leaks can occur in the sprinkler head or the water supply lines.
Clogged sprinkler head: A clogged sprinkler head can reduce the effectiveness of the sprinkler head.
Worn parts: Worn parts, such as the sprinkler head or gears, can reduce the effectiveness of the sprinkler head.

Q: How do I troubleshoot a rotating sprinkler head?

A: To troubleshoot a rotating sprinkler head, follow these steps:

Check the water supply: Check the water supply to ensure it is turned on and functioning properly.
Check the sprinkler head: Check the sprinkler head to ensure it is properly aligned and set to the correct angle.
Check for leaks: Check for leaks in the sprinkler head or the water supply lines.

Conclusion

Rotating sprinkler heads are a common feature in many residential and commercial irrigation systems. By understanding how they work and how to choose the right one for your needs, you can ensure efficient and effective watering of your lawn, garden, or other outdoor space.