Calculating Sprinkler Water Coverage Area A Step-by-Step Guide
Hey guys! In this article, we're diving into a fun and practical math problem: calculating the area covered by a rotating sprinkler head. This is a great example of how geometry concepts, like circles and sectors, can be applied in real-world scenarios. Whether you're a student tackling homework, a homeowner planning your lawn care, or just a math enthusiast, this breakdown will help you understand the steps involved in finding the solution. We'll break down the problem step by step, making it super easy to follow along. So, let's get started and explore how to calculate the watered area! Remember, math can be both useful and engaging when we connect it to everyday situations.
Understanding the Problem
The key here is to understand that the sprinkler's spray creates a sector of a circle. Think of a pizza slice – that's essentially what we're dealing with! The sprinkler head is at the center of the circle, the water spray reaches a certain radius (the distance the water travels), and the angle of rotation determines the size of the slice (or sector) that gets watered.
In this specific problem, we're given two crucial pieces of information:
- The radius of the circle: This is the distance the water sprays, which is 20 feet.
- The central angle of the sector: This is the angle covered by the sprinkler's rotation, which is 80 degrees.
Our mission is to find the area of this sector. This represents the area of the grass that gets watered by the sprinkler. To do this, we'll need to recall the formula for the area of a sector and apply it to the given values. So, let's move on to the next step: understanding the formula.
Breaking Down the Information
Let's reiterate the critical information provided in the problem. First and foremost, we know that the rotating sprinkler head sprays water as far as 20 feet. This is extremely important because it gives us the radius of the circle that the sprinkler's spray covers. Think of it like drawing a circle with a compass; the 20 feet is the length of the compass arm, determining the circle's size. Understanding this radius is crucial for calculating the area, as it's a direct input into our sector area formula.
Secondly, the problem states that the head is set to cover a central angle of 80 degrees. Now, imagine that pizza slice again. The central angle is the angle formed at the tip of the slice, where all the edges meet. In our sprinkler scenario, it’s the angle the sprinkler rotates through as it sprays water. This angle dictates what fraction of the whole circle the sprinkler is actually covering. If the angle were 360 degrees, the sprinkler would cover the entire circle. But with 80 degrees, it covers just a portion. Thus, knowing the central angle is essential for determining the sector's area accurately.
By carefully identifying these two key pieces of information – the radius and the central angle – we've laid a solid foundation for solving the problem. We now understand the dimensions of the area we need to calculate, setting us up perfectly for applying the correct formula.
The Formula for the Area of a Sector
The area of a sector is a fraction of the area of the entire circle. The fraction is determined by the central angle of the sector compared to the total angle of a circle (360 degrees).
The formula is:
Area of Sector = (Central Angle / 360°) * π * radius²
Let's break down this formula:
- Central Angle: This is the angle of the sector, given in degrees.
- 360°: This represents the total degrees in a circle.
- π (pi): This is a mathematical constant, approximately equal to 3.14159.
- radius²: This is the radius of the circle squared.
Basically, what this formula does is calculate the area of the full circle (π * radius²) and then multiplies it by the fraction of the circle represented by the sector (Central Angle / 360°). It's a pretty straightforward concept when you break it down like this!
Understanding the Components
To truly master calculating the area of a sector, it's vital that you deeply comprehend each component of the formula. Let's start with the central angle. This angle, measured in degrees, is the cornerstone of our calculation. It defines the 'slice' or portion of the circle we're interested in. A larger central angle means a bigger slice, and consequently, a larger sector area. Think of it like adjusting the nozzle on a sprinkler to cover a wider or narrower area – that adjustment directly correlates to the central angle.
Next, we have 360°, which represents the total degrees in a complete circle. This number serves as our baseline, our 'whole' against which we compare the central angle. By dividing the central angle by 360°, we're essentially finding the fraction of the circle that the sector occupies. This fraction is crucial because it tells us what proportion of the entire circle's area our sector will cover.
Then comes π (pi), the famous mathematical constant approximately equal to 3.14159. Pi is the ratio of a circle's circumference to its diameter, and it's fundamental to any circle-related calculation. It's an irrational number, meaning its decimal representation goes on forever without repeating, but for most practical purposes, we use the approximation 3.14159 or even just 3.14. Pi is what links the radius of a circle to its area, making it an indispensable part of our formula.
Finally, we have radius², which is the radius of the circle squared (multiplied by itself). The radius, as we discussed earlier, is the distance from the center of the circle to any point on its edge. Squaring it gives us a measure proportional to the area. A larger radius naturally leads to a much larger area, as the area increases with the square of the radius. This is why the radius plays such a significant role in the sector area formula.
By grasping the meaning and function of each of these components – the central angle, 360°, π, and the radius squared – you'll gain a much stronger understanding of how the sector area formula works and how to apply it effectively.
Applying the Formula to the Problem
Now for the fun part – let's plug in the values we know into the formula and solve for the area!
We have:
- Central Angle = 80°
- Radius = 20 feet
So, our formula becomes:
Area of Sector = (80° / 360°) * π * (20 feet)²
First, let's simplify the fraction 80° / 360°. Both 80 and 360 are divisible by 40, so we can reduce the fraction to 2/9.
Next, let's calculate (20 feet)² which is 20 * 20 = 400 square feet.
Now, we have:
Area of Sector = (2/9) * π * 400 square feet
Multiply 2/9 by 400 to get 800/9.
So, the area of the sector is:
Area of Sector = (800/9) * π square feet
Step-by-Step Calculation
To ensure we're all on the same page, let's meticulously walk through the calculation process step by step. This will help solidify your understanding and prevent any potential errors.
Step 1: Substitute the known values into the formula.
We start with our formula: Area of Sector = (Central Angle / 360°) * π * radius². We know the central angle is 80° and the radius is 20 feet. Substituting these values, we get:
Area of Sector = (80° / 360°) * π * (20 feet)²
This step is all about careful replacement, making sure each value goes in the correct spot.
Step 2: Simplify the fraction.
The fraction 80° / 360° can be simplified. Both 80 and 360 share a common factor of 40. Dividing both the numerator and denominator by 40, we get:
80° / 360° = 2/9
Simplifying fractions makes the subsequent calculations easier.
Step 3: Calculate the square of the radius.
We need to calculate (20 feet)², which means 20 feet multiplied by itself:
(20 feet)² = 20 feet * 20 feet = 400 square feet
Remember, when we square a length, we get an area, hence the square feet unit.
Step 4: Plug the simplified values back into the formula.
Now we substitute our simplified values back into the formula:
Area of Sector = (2/9) * π * 400 square feet
We're getting closer to our final answer!
Step 5: Multiply the fraction and the area.
Next, we multiply the fraction 2/9 by 400 square feet:
(2/9) * 400 square feet = 800/9 square feet
This step combines the fraction representing the sector's proportion with the total area influenced by the radius.
Step 6: Write the final answer.
Finally, we have our area of the sector:
Area of Sector = (800/9) * π square feet
This is the area of the grass watered by the sprinkler. We leave the answer in terms of π for exactness, unless we're asked to approximate it using a decimal value for π.
By following these steps meticulously, you can confidently calculate the area of any sector, given its central angle and radius.
The Correct Answer
Looking at the answer choices provided, we need to match our result, (800/9) * π square feet, with one of the options. The correct answer is:
- A. (800/9) * π ft²
Confirming the Solution
To ensure we've arrived at the correct solution, it's always a good idea to take a moment and confirm our answer. We can do this by thinking logically about the problem and the steps we've taken.
First, let's consider the reasonableness of our answer. We calculated the area of the sector to be (800/9)π square feet. Since π is approximately 3.14, this gives us an area a little less than 300 square feet. This seems reasonable, considering the sprinkler sprays water up to 20 feet, but only covers a portion of a full circle (80 degrees out of 360).
Next, we can retrace our steps to check for any potential errors. We started by correctly identifying the radius and central angle from the problem statement. We then applied the sector area formula accurately, substituting the values and simplifying the fraction. The squaring of the radius and the final multiplication were also performed correctly. By systematically reviewing each step, we can increase our confidence in the solution.
Another helpful way to confirm our solution is to compare it to other possible answers. The problem provided answer choices, and our calculated answer matches one of them perfectly. This adds further support to our solution.
Finally, we can think about the units. Our answer is in square feet, which is the correct unit for measuring area. If we had ended up with feet or cubic feet, we would know something had gone wrong.
By taking these confirmation steps, we can be confident that (800/9)π square feet is indeed the correct area of grass watered by the sprinkler. This process of verification is a crucial part of problem-solving, helping to avoid mistakes and build confidence in our mathematical abilities.
So there you have it, guys! We successfully calculated the area of grass watered by a rotating sprinkler head. By understanding the concept of a sector, the formula for its area, and applying it step-by-step, we were able to solve the problem. This is a great example of how math can be applied to everyday situations. I hope this explanation was helpful and made the process clear and understandable. Keep practicing, and you'll be a math whiz in no time!
