Math Problem Statement

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.)

Solution

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=400r = 400 feet and a central angle of θ=220\theta = 220^\circ.

The area AA of a sector of a circle can be calculated using the formula:

A=12r2θA = \frac{1}{2} r^2 \theta

where:

  • rr is the radius of the circle,
  • θ\theta is the central angle in radians.

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

θradians=θdegrees×π180\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}

So,

θradians=220×π180=220π180=11π9 radians\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=12×(400)2×11π9A = \frac{1}{2} \times (400)^2 \times \frac{11\pi}{9} A=12×160000×11π9A = \frac{1}{2} \times 160000 \times \frac{11\pi}{9} A=80000×11π980000×3.8397307176 square feetA = 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.

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Math Problem Analysis

Mathematical Concepts

Geometry
Circle Sector Area

Formulas

Area of a sector of a circle

Theorems

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Suitable Grade Level

High School