An irrigation system uses a straight sprinkler pipe r = 400 ft long that pivots around a central point as shown. Because of an obstacle the pipe is allowed to pivot through 𝜃 = 220° only. Find the area irrigated by this system. (Round your answer to the nearest whole number.)

To find the area irrigated by the system, we can think of the system as creating a sector of a circle with a radius of \( r = 400 \) feet and a central angle of \( \theta = 220^\circ \).

The area \( A \) of a sector of a circle can be calculated using the formula:

\[
A = \frac{1}{2} r^2 \theta
\]

where:
- \( r \) is the radius of the circle,
- \( \theta \) is the central angle in radians.

First, we need to convert the angle from degrees to radians. The conversion factor is:

\[
\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}
\]

So,

\[
\theta_{\text{radians}} = 220^\circ \times \frac{\pi}{180} = \frac{220\pi}{180} = \frac{11\pi}{9} \text{ radians}
\]

Now, we can calculate the area:

\[
A = \frac{1}{2} \times (400)^2 \times \frac{11\pi}{9}
\]

\[
A = \frac{1}{2} \times 160000 \times \frac{11\pi}{9}
\]

\[
A = 80000 \times \frac{11\pi}{9} \approx 80000 \times 3.8397 \approx 307176 \text{ square feet}
\]

Rounding to the nearest whole number, the area irrigated by the system is approximately **307,176 square feet**.

Would you like further details on any part of the solution, or do you have any questions?

Here are 5 related questions for further exploration:
1. What would the area be if the sprinkler pivoted through a full 360°?
2. How does the area change if the radius of the sprinkler pipe is doubled?
3. What would the irrigated area be if the angle were reduced to 180°?
4. How would the area change if the radius was increased by 50 feet?
5. How do you convert radians back to degrees?

**Tip:** Always ensure to convert angles to radians when using them in trigonometric or sector area formulas.

Discover how to calculate the area irrigated by a sprinkler system with a 220° pivot. This math problem involves finding the area of a sector of a circle with a given radius and central angle. Learn step-by-step with detailed calculations and formulas. Suitable for high school students studying geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation