Calculating Sprinkler Coverage Area A Step-by-Step Guide

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In this article, we will explore how to calculate the area of grass watered by a rotating sprinkler head. This is a practical application of geometry and trigonometry, specifically dealing with the area of a sector of a circle. Understanding these calculations can be beneficial for various real-world scenarios, such as landscape design, irrigation planning, and even understanding the mechanics of rotating devices. This comprehensive guide will walk you through the problem step-by-step, ensuring you grasp the underlying concepts and can apply them to similar situations. We will delve into the formula for the area of a sector, discuss how to convert angles from degrees to radians, and provide a detailed explanation of the calculations involved. By the end of this article, you will have a solid understanding of how to determine the coverage area of a rotating sprinkler and be able to tackle related problems with confidence.

We are given that a rotating sprinkler head sprays water as far as 20 feet. This distance represents the radius of the circular area the sprinkler can cover. The sprinkler head is set to cover a central angle of 80 degrees. Our task is to determine the area of the grass that will be watered by this sprinkler setting. This problem requires us to calculate the area of a sector of a circle, which is a portion of the circle defined by the central angle. The central angle, in this case, is the angle formed at the center of the circle (where the sprinkler head is located) by the two extreme positions of the water spray. The area of the watered grass corresponds to the area of this sector. To solve this, we'll use the formula for the area of a sector, which involves the radius of the circle and the central angle. We will need to ensure that the angle is in the correct units (radians) before applying the formula. The following sections will break down the steps involved in calculating this area, providing a clear and concise solution to the problem.

Before diving into the calculations, let's solidify our understanding of the key concepts involved. The primary concept here is the area of a sector of a circle. A sector is a pie-shaped portion of a circle, bounded by two radii and the intercepted arc. Imagine slicing a pizza; each slice is a sector of the circular pizza. The area of this sector depends on two factors: the radius of the circle and the central angle that defines the sector. The radius is the distance from the center of the circle to any point on its circumference, which in our case is the distance the sprinkler sprays water (20 feet). The central angle is the angle formed at the center of the circle by the two radii that bound the sector, given as 80 degrees in this problem. The formula to calculate the area of a sector is given by:

Area of sector = (θ / 360°) × πr²

Where:

  • θ is the central angle in degrees,
  • r is the radius of the circle.

Another crucial concept is the conversion between degrees and radians. Radians are an alternative unit for measuring angles, often used in mathematical and scientific contexts. One full circle is 360 degrees, which is equivalent to 2Ï€ radians. To convert degrees to radians, we use the following conversion factor:

Radians = (Degrees × π) / 180

In some cases, using radians in the sector area formula can simplify calculations, especially in more advanced mathematical contexts. Understanding these concepts thoroughly will enable you to approach this and similar problems with greater confidence and accuracy. The next sections will apply these concepts to solve the problem at hand, demonstrating the practical application of these principles.

To calculate the area of the grass watered by the sprinkler, we'll follow these steps:

  1. Identify the given values: The radius of the circle (r) is 20 feet, and the central angle (θ) is 80 degrees.
  2. Apply the formula for the area of a sector: The formula is Area = (θ / 360°) × πr².
  3. Substitute the given values into the formula: Area = (80° / 360°) × π × (20 ft)².
  4. Simplify the expression:
    • Area = (2/9) × Ï€ × 400 sq ft
    • Area = (800/9)Ï€ sq ft

Therefore, the area of the grass that will be watered is (800/9)π square feet. This calculation directly applies the formula for the area of a sector, utilizing the given radius and central angle. The simplification steps involve reducing the fraction representing the proportion of the circle covered by the sector and then multiplying by the square of the radius and π. This result gives us the area in terms of π, which is a common way to express such answers in mathematics to maintain precision. If a numerical approximation is desired, one could substitute π with its approximate value (approximately 3.14159) and calculate the result. The next section will discuss the final answer and its implications, as well as provide some context for similar problems.

The calculated area of the grass that will be watered by the rotating sprinkler is (800/9)π square feet. This is the exact answer, expressed in terms of π. If we need a numerical approximation, we can substitute π with approximately 3.14159 to get an approximate area of 279.25 square feet. This result gives us a clear understanding of the sprinkler's coverage area when set to spray water at a central angle of 80 degrees and a radius of 20 feet. Understanding the coverage area is crucial for efficient irrigation, ensuring that the grass receives adequate water without wastage. Overwatering can lead to waterlogging, while underwatering can cause the grass to dry out. By accurately calculating the coverage area, we can optimize the sprinkler settings to achieve the best watering results.

This type of calculation has implications beyond just sprinkler systems. It can be applied in various fields, such as:

  • Landscape Design: Determining the area covered by landscaping elements.
  • Engineering: Calculating the surface area covered by rotating machinery.
  • Mathematics Education: Reinforcing the concepts of geometry and trigonometry.

The ability to calculate the area of a sector is a valuable skill that can be applied in many practical scenarios. By mastering this concept, you can solve a wide range of problems involving circular areas and sectors. The next section will provide some practice questions to further solidify your understanding of the material.

To reinforce your understanding of calculating the area of a sector, here are a few practice questions:

  1. A sprinkler head sprays water as far as 15 feet and is set to cover a central angle of 120 degrees. What area of grass will be watered?
  2. A rotating spotlight covers a distance of 30 meters and is set to illuminate an arc of 60 degrees. What is the area illuminated by the spotlight?
  3. A pizza slice has a central angle of 45 degrees and a radius of 8 inches. What is the area of the pizza slice?

These questions are designed to help you apply the concepts and formulas discussed in this article. Try solving them on your own, and then compare your answers with the solutions provided below. Working through these practice questions will enhance your problem-solving skills and solidify your understanding of the area of a sector.

Here are the solutions to the practice questions provided above:

  1. Sprinkler Question:
    • Given: Radius (r) = 15 feet, Central Angle (θ) = 120 degrees
    • Area = (θ / 360°) × Ï€r²
    • Area = (120° / 360°) × Ï€ × (15 ft)²
    • Area = (1/3) × Ï€ × 225 sq ft
    • Area = 75Ï€ sq ft
    • Answer: The area of grass watered is 75Ï€ square feet.
  2. Spotlight Question:
    • Given: Radius (r) = 30 meters, Central Angle (θ) = 60 degrees
    • Area = (θ / 360°) × Ï€r²
    • Area = (60° / 360°) × Ï€ × (30 m)²
    • Area = (1/6) × Ï€ × 900 sq m
    • Area = 150Ï€ sq m
    • Answer: The area illuminated by the spotlight is 150Ï€ square meters.
  3. Pizza Slice Question:
    • Given: Radius (r) = 8 inches, Central Angle (θ) = 45 degrees
    • Area = (θ / 360°) × Ï€r²
    • Area = (45° / 360°) × Ï€ × (8 in)²
    • Area = (1/8) × Ï€ × 64 sq in
    • Area = 8Ï€ sq in
    • Answer: The area of the pizza slice is 8Ï€ square inches.

By reviewing these solutions, you can check your work and gain further insight into the problem-solving process. If you encountered any difficulties, revisit the step-by-step solution and explanations provided earlier in this article. The next section will summarize the key points discussed and offer some final thoughts on the topic.

In this article, we've explored the process of calculating the area of grass watered by a rotating sprinkler head. We started by understanding the problem statement, which involved determining the area of a sector of a circle. We then delved into the key concepts, including the formula for the area of a sector and the conversion between degrees and radians. We followed a step-by-step solution to calculate the area, which involved identifying the given values, applying the formula, and simplifying the expression. The final answer, (800/9)Ï€ square feet, represents the area of the grass watered by the sprinkler. We also discussed the implications of this calculation in various fields, such as landscape design and engineering. To reinforce your understanding, we provided practice questions and their solutions, allowing you to apply the concepts and formulas learned. By mastering the calculation of the area of a sector, you can solve a wide range of practical problems involving circular areas and sectors. This skill is valuable in many real-world scenarios, from optimizing irrigation systems to designing landscapes. We hope this comprehensive guide has provided you with a clear and concise understanding of how to calculate the area of a sector and its practical applications.