Select The Correct Answer.A Rotating Sprinkler Head Sprays Water As Far As 20 Feet. The Head Is Set To Cover A Central Angle Of 80 ∘ 80^{\circ} 8 0 ∘ . What Area Of Grass Will Be Watered?A. 30 5 Π Ft 2 \frac{30}{5} \pi \text{ Ft}^2 5 30 ​ Π Ft 2 B. $\frac{90}{5}

Introduction

Rotating sprinkler heads are a common sight in many gardens and lawns, providing a convenient and efficient way to water large areas of grass. However, to ensure that the sprinkler head is set to water the desired area, it's essential to understand the relationship between the sprinkler's angle of rotation and the area it covers. In this article, we'll explore how to calculate the area of grass that will be watered by a rotating sprinkler head.

Understanding the Problem

The problem states that a rotating sprinkler head sprays water as far as 20 feet and is set to cover a central angle of $80^{\circ}$. We need to find the area of grass that will be watered by this sprinkler head.

Calculating the Area

To calculate the area of grass that will be watered, 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, and $r$ is the radius of the circle.

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

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

Simplifying the expression, we get:

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

$$A = \frac{800}{9} \pi$$

Evaluating the Answer Choices

Now that we have calculated the area of grass that will be watered, let's evaluate the answer choices:

A. $\frac{30}{5} \pi \text{ ft}^2$

B. $\frac{90}{5} \pi \text{ ft}^2$

C. $\frac{800}{9} \pi \text{ ft}^2$

D. $\frac{900}{9} \pi \text{ ft}^2$

Comparing our calculated answer with the answer choices, we can see that the correct answer is:

C. $\frac{800}{9} \pi \text{ ft}^2$

Conclusion

In this article, we calculated the area of grass that will be watered by a rotating sprinkler head with a central angle of $80^{\circ}$ and a radius of 20 feet. We used the formula for the area of a sector of a circle to find the area, and then evaluated the answer choices to determine the correct answer. The correct answer is C. $\frac{800}{9} \pi \text{ ft}^2$.

Final Answer

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

Additional Information

• The formula for the area of a sector of a circle is $A = \frac{\theta}{360} \pi r^2$, where $A$ is the area of the sector, $\theta$ is the central angle in degrees, and $r$ is the radius of the circle.
• The central angle is the angle between the two radii that form the sector.
• The radius is the distance from the center of the circle to the edge of the sector.

References

• [1] "Area of a Sector of a Circle". Math Open Reference. Retrieved 2023-02-20.
• [2] "Rotating Sprinkler Head". Wikipedia. Retrieved 2023-02-20.
  Frequently Asked Questions (FAQs) about Rotating Sprinkler Heads ====================================================================

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

A: The primary purpose of a rotating sprinkler head is to water a large area of grass efficiently. It sprays water in a circular pattern, covering a significant amount of ground with each rotation.

Q: How does a rotating sprinkler head work?

A: A rotating sprinkler head works by using a central axis to rotate a series of nozzles or sprinkler arms. As the head rotates, the nozzles or sprinkler arms spray water in a circular pattern, covering the desired area.

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 cover a large area of grass with each rotation, reducing the time and effort required to water the lawn.
• Uniform watering: The circular pattern of the sprinkler head ensures that the water is distributed evenly across the lawn, reducing the risk of overwatering or underwatering.
• Reduced water waste: By using a rotating sprinkler head, you can reduce the amount of water wasted due to evaporation or runoff.

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. Larger lawns require more powerful sprinkler heads.
• Water pressure: Make sure the sprinkler head is compatible with your water pressure. Some sprinkler heads require high water pressure to function properly.
• Sprinkler head type: There are different types of sprinkler heads available, including impact sprinklers, gear-driven sprinklers, and pop-up sprinklers. Choose the type that best suits your needs.

Q: How do I install a rotating sprinkler head?

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

1. Choose a suitable location for the sprinkler head, taking into account the size of your lawn and the water pressure.
2. Dig a hole for the sprinkler head, making sure it is deep enough to accommodate the head and any necessary connections.
3. Connect the sprinkler head to the water supply, following the manufacturer's instructions.
4. Adjust the sprinkler head to the desired angle and height.
5. Test the sprinkler head to ensure it is working properly.

Q: How do I maintain a rotating sprinkler head?

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

1. Regularly inspect the sprinkler head for damage or wear.
2. Clean the sprinkler head regularly to ensure it is functioning properly.
3. Adjust the sprinkler head as needed to ensure it is watering the lawn evenly.
4. Replace the sprinkler head if it becomes damaged or worn out.

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

A: Some common problems with rotating sprinkler heads include:

• Clogged nozzles or sprinkler arms
• Incorrectly adjusted sprinkler head
• Insufficient water pressure
• Damaged or worn-out sprinkler head

Q: How do I troubleshoot common problems with rotating sprinkler heads?

A: To troubleshoot common problems with rotating sprinkler heads, follow these steps:

1. Check the nozzles or sprinkler arms for clogs or blockages.
2. Adjust the sprinkler head to ensure it is watering the lawn evenly.
3. Check the water pressure to ensure it is sufficient.
4. Replace the sprinkler head if it becomes damaged or worn out.

Conclusion

Rotating sprinkler heads are a convenient and efficient way to water large areas of grass. By understanding how they work and how to choose the right one for your lawn, you can ensure that your lawn is watered evenly and efficiently. Regular maintenance and troubleshooting can also help to extend the life of your sprinkler head and prevent common problems.