Problem 76 - Algebra and Trigonometry

Question

Irrigation 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.

Step-by-Step Solution

Answer

The area irrigated is approximately 219,911.5 square feet.

Step 1: Understanding the problem setup

The sprinkler system consists of a straight pipe with a length of 300 feet that pivots around a central point. Due to obstacles, the pipe can only pivot through an angle of 280 degrees.

Step 2: Calculating the area of the sector

The sprinkler forms a sector of a circle. The formula for the area of a sector is \( A = \frac{\theta}{360} \times \pi \times r^2 \), where \( \theta \) is the angle in degrees and \( r \) is the radius of the circle. In this case, \( \theta = 280 \) degrees and \( r = 300 \) ft.

Step 3: Applying the sector area formula

Substitute \( \theta = 280 \) and \( r = 300 \) into the formula: \( A = \frac{280}{360} \times \pi \times (300)^2 \).

Step 4: Simplifying the expression

\( A = \frac{280}{360} \times \pi \times 90000 \). Simplifying the fraction \( \frac{280}{360} \) gives \( \frac{7}{9} \).

Step 5: Final Calculation

\( A = \frac{7}{9} \times \pi \times 90000 = \frac{630000 \pi}{9} \). Calculate this to find the area: \( 70000 \pi \ \approx 219911.5 \ \text{square feet} \).

Key Concepts

Circle Geometry

Circle geometry is an intriguing field of mathematics focusing on properties and relations of circles and shapes derived from them. A central concept in circle geometry is the circle itself, which is a set of points equidistant from a center point. A useful associated term is the radius, which measures that distance. In many real-world applications, like the sprinkler system in the irrigation problem, understanding circle geometry is crucial.

When a circular section is created by two radii and an intercepted arc, we call this a sector of a circle. The equation for calculating the area of a sector—an essential part of circle geometry—is:

• Area (A) = \( \frac{\theta}{360} \times \pi \times r^2 \)

Where \( \theta \) is the central angle of the sector in degrees, and \( r \) is the radius of the circle. If you understand this formula, you can solve many practical circle geometry problems with ease.

Irrigation System

Irrigation systems are crucial in agriculture. They provide water to crops when rainfall is insufficient. The system described in the exercise uses a rotating sprinkler that waters fields arranged in a circle pattern. Imagine this setup: a long pipe acting as an arm pivots around a central point, creating a sector of circle on the field.

In our example, the field is watered in an arc with a specific angle due to an obstacle, such as a building or tree. Though the pipe's full circular motion is restricted, understanding how to calculate the area effectively covered remains essential. By using the circle sector area formula, you can determine the exact footprint of water coverage and thereby evaluate the system efficiency.

Already, agricultural engineers rely on concepts from circle geometry to optimize the size and configuration of fields and irrigation systems.

Angle Measurement

Angle measurement is fundamental when dealing with geometrical shapes, like the ones encountered in circle geometry. The specific case of the irrigation system requires understanding how to measure and appropriately manipulate angles.

Angles are typically measured in degrees, where a full circle encompasses 360 degrees. In the exercise problem, the sprinkler pivots through 280 degrees, forming the angle of the sector. This explains the motion range of the irrigation pipe within its operational field.

Measuring angles correctly influences the calculation of areas or lengths in circular geometry heavily. For the sprinkler:

• The sector angle is central to modeling the area covered.
• Any deviations in angle measurement could lead to errors in estimating the overall irrigation area.

Developing a solid foundation in angle measurement and its application helps accurately solve problems in both classroom scenarios and practical applications.