Problem 76 - Precalculus Mathematics for Calculus

Question

An irrigation system uses a straight sprinkler pipe \(300 \mathrm{ft}\) long that pivots around a central point as shown. Due to an obstacle the pipe is allowed to pivot through \(280^{\circ}\) only. Find the area irrigated by this system. (IMAGE CAN NOT COPY)

Step-by-Step Solution

Answer

The area irrigated is approximately 219,911 square feet.

Step 1: Understanding the Problem

The sprinkler system is in the shape of a circle segment. The central pivot point limits the circle to a sector with an angle of \(280^{\circ}\). We need to calculate the area of this sector using the formula for the area of a sector of a circle.

Step 2: Formula for Area of a Sector

The formula to find the area \(A\) of a sector with central angle \(\theta\) (in degrees) and radius \(r\) is \(A = \frac{\theta}{360} \times \pi r^2\). So, we need to apply this formula with our given angle and radius.

Step 3: Convert Degrees and Identify Values

Here, \(\theta = 280^{\circ}\) and the radius \(r = 300 \text{ ft}\). Ensure that the angle is correctly identified in degrees since it is already provided in degrees.

Step 4: Calculate Area of the Sector

Substitute \(\theta = 280\) and \(r = 300\) into the sector area formula: \[A = \frac{280}{360} \times \pi \times (300)^2\]Calculate the area by simplifying the fraction and solving the equation.

Step 5: Perform Calculation

First, simplify the fraction:\(\frac{280}{360} = \frac{7}{9}\).Substitute back in:\[A = \frac{7}{9} \times \pi \times 90000\]Simplify and calculate:\[A = \frac{7}{9} \times 282743.34 \approx 219911.45 \text{ square feet}\]

Step 6: Round Off the Area

For practical purposes, round the area to a sensible degree of accuracy:\[A \approx 219,911 \text{ square feet}\]

Key Concepts

irrigation system

An irrigation system is a method of supplying water to areas of land to grow crops, maintain landscapes, and rehydrate soil. In agricultural settings, irrigation is critical where rainfall is insufficient to support sustained plant growth. A common irrigation setup is the use of a sprinkler system, which distributes water in a controlled manner.

Such systems often incorporate components that are strategically aligned to cover as much area as possible, ensuring efficient water use. The sprinkler system described in the exercise is a type commonly used in fields known as a pivot irrigation system. Here’s how it works:

• The central pivot is fixed, allowing the attached pipe to rotate.
• Water is distributed evenly as the pipe rotates, covering a circular area or a sector of a circle.
• The length of the pipe determines how far the water reaches, simulating a radius in the circular motion.
• Obstacles or land boundaries can restrict the movement angle, like in the given problem where the range is limited to 280°.

The goal is to maximize water coverage without waste, making understanding geometric principles crucial in planning and optimizing irrigation systems.

circular sector

A circular sector is a portion of a circle resembling a 'slice of pie'. It is defined by a central angle and the two radii that make up the sector's edge. In geometry, sectors are important for calculating areas used in various practical applications like the irrigation example.

To better understand, consider these characteristics of a circular sector:

• Central Angle: It is the angle subtended at the center, determining the "openness" of the sector.
• Radius: The distance from the center to the boundary or arc of the circle.
• Arc Length: The distance along the curved edge of the sector.
• Area of the Sector: Calculated using the formula \(A = \frac{\theta}{360} \times \pi r^2\).

The sector’s area gives us a perspective on how much space within a circle is utilized. This is vital in situations like the irrigation system, where you need to know the exact footprint of water distribution. Sectors allow us to adjust parameters easily, like radius (length of the sprinkler pipe) and angle (the pivot range), impacting the overall area covered.

angle measurement

Angle measurement is fundamental when dealing with circular segments, as it directs their scope and area coverage. In mathematics, angles are measured in degrees or radians, providing us a means to comprehend the extent of bending or rotation.

To translate angle concepts effectively:

• Degrees: Often used in practical applications, a full circle is 360°, making it easy to divide portions like sectors.
• Radians: The "natural" unit of angular measurement derived from the radius of the circle. Helpful in calculus and theoretical contexts.
• Conversion: Knowing how to switch between degrees and radians is useful: \(1 ext{ radian} = \frac{180}{\pi} ext{ degrees}\).

In our irrigation example, the angle of 280° signifies that the pipe pivots to cover nearly 78% of a full circular orbit. Understanding this allows us to apply formulas efficiently, adjust system parameters, and ensure optimal land coverage.

Grasping angle measurement, therefore, is not just about numbers but about applying them to real-world systems to optimize their performance.