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 $80^{\circ}$. What Area Of Grass Will Be Watered?A. 200 9 Π Ft 2 \frac{200}{9} \pi \, \text{ft}^2 9 200 ​ Π Ft 2 B.

Introduction

When it comes to maintaining a lush and healthy lawn, proper watering is essential. A rotating sprinkler head is a common tool used to water large areas of grass efficiently. However, to ensure that the sprinkler head is covering the desired area, it's crucial to understand the relationship between the sprinkler's angle and the area it covers. In this article, we'll delve into the mathematics behind calculating the area of grass watered by a rotating sprinkler head.

Understanding the Problem

Given 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 determine the area of grass that will be watered. To solve this problem, we'll use the concept of sectors and the formula for calculating the area of a sector of a circle.

The Formula for the Area of a Sector

The area of a sector of a circle can be calculated using the formula:

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

where:

• $A$ is the area of the sector
• $\theta$ is the central angle in degrees
• $r$ is the radius of the circle

Applying the Formula to the Problem

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:

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

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

However, we need to consider that the sprinkler head is rotating, and the area it covers is a fraction of the total area of the circle. To find the correct answer, we need to multiply the area of the sector by the fraction of the circle that the sprinkler head covers.

Calculating the Fraction of the Circle Covered

Since the sprinkler head is set to cover a central angle of $80^{\circ}$, it covers a fraction of the circle equal to:

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

Finding the Correct Answer

Multiplying the area of the sector by the fraction of the circle covered, we get:

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

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

However, we need to consider that the sprinkler head is rotating, and the area it covers is a fraction of the total area of the circle. To find the correct answer, we need to multiply the area of the sector by the fraction of the circle that the sprinkler head covers.

Conclusion

In conclusion, the area of grass watered by a rotating sprinkler head can be calculated using the formula for the area of a sector of a circle. By applying the formula and considering the fraction of the circle covered by the sprinkler head, we can determine the correct answer.

The Correct Answer

The correct answer is:

$$\frac{200}{9} \pi \, \text{ft}^2$$

This answer takes into account the fraction of the circle covered by the sprinkler head and provides the correct area of grass that will be watered.

Discussion

What do you think about the calculation of the area of grass watered by a rotating sprinkler head? Do you have any questions or concerns about the formula or the solution? Share your thoughts in the comments below!

Related Topics

• Calculating the area of a circle
• Understanding the concept of sectors
• Applying formulas to real-world problems

References

• [1] "Mathematics for the Nonmathematician" by Morris Kline
• [2] "Calculus" by Michael Spivak
• [3] "Geometry" by Euclid

Additional Resources

• Khan Academy: Calculus
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Calculus and Geometry
  Q&A: Calculating the Area of Grass Watered by a Rotating Sprinkler Head ====================================================================

Introduction

In our previous article, we explored the mathematics behind calculating the area of grass watered by a rotating sprinkler head. We used the formula for the area of a sector of a circle to determine the correct answer. In this article, we'll answer some frequently asked questions about the topic.

Q: What is the formula for the area of a sector of a circle?

A: 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
• $r$ is the radius of the circle

Q: How do I calculate the area of grass watered by a rotating sprinkler head?

A: To calculate the area of grass watered by a rotating sprinkler head, you need to follow these steps:

1. Determine the central angle of the sprinkler head in degrees.
2. Determine the radius of the sprinkler head in feet.
3. Plug the values into the formula for the area of a sector of a circle.
4. Simplify the expression to find the area of the sector.
5. Multiply the area of the sector by the fraction of the circle covered by the sprinkler head.

Q: What is the fraction of the circle covered by a rotating sprinkler head?

A: The fraction of the circle covered by a rotating sprinkler head is equal to the central angle of the sprinkler head divided by 360.

Q: How do I determine the central angle of a rotating sprinkler head?

A: The central angle of a rotating sprinkler head is usually specified by the manufacturer. If you're unsure, you can measure the angle using a protractor or a angle-measuring tool.

Q: What if the sprinkler head is not rotating at a constant speed?

A: If the sprinkler head is not rotating at a constant speed, you'll need to take into account the time it takes for the sprinkler head to complete one rotation. You can do this by multiplying the area of the sector by the fraction of the time the sprinkler head is rotating.

Q: Can I use this formula to calculate the area of grass watered by a sprinkler system with multiple sprinkler heads?

A: Yes, you can use this formula to calculate the area of grass watered by a sprinkler system with multiple sprinkler heads. Simply add up the areas of each sector and multiply by the fraction of the circle covered by each sprinkler head.

Q: What are some common mistakes to avoid when calculating the area of grass watered by a rotating sprinkler head?

A: Some common mistakes to avoid when calculating the area of grass watered by a rotating sprinkler head include:

• Failing to account for the fraction of the circle covered by the sprinkler head
• Using the wrong formula or values
• Not considering the time it takes for the sprinkler head to complete one rotation
• Not taking into account the shape and size of the lawn

Conclusion

Calculating the area of grass watered by a rotating sprinkler head is a simple process that requires the use of the formula for the area of a sector of a circle. By following the steps outlined in this article, you can determine the correct answer and ensure that your lawn is receiving the right amount of water.

Additional Resources

• Khan Academy: Calculus
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Calculus and Geometry
• Sprinkler system manufacturers' websites for more information on sprinkler head specifications and calculations.

Related Topics

• Calculating the area of a circle
• Understanding the concept of sectors
• Applying formulas to real-world problems

References

• [1] "Mathematics for the Nonmathematician" by Morris Kline
• [2] "Calculus" by Michael Spivak
• [3] "Geometry" by Euclid