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Difficulty: HardCircles

A radar system on a ship sweeps a 45∘45^\circ sector of the surrounding water. If the total area scanned by this single sweep is 32π32\pi square nautical miles, what is the maximum detection range (radius) of the radar in nautical miles?

  1. A
    2
  2. B
    32
  3. 16Answer
  4. D
    128

Answer

The maximum detection range (radius) of the radar is 16 nautical miles.
The area of a sector is found using the formula A=θ360πr2A = \frac{\theta}{360} \pi r^2. Substituting the given values gives 32π=45360πr232\pi = \frac{45}{360} \pi r^2. Simplifying the fraction 45360\frac{45}{360} to 18\frac{1}{8} and dividing both sides by π\pi yields 32=18r232 = \frac{1}{8} r^2. Multiplying both sides by 8 isolates r2r^2, resulting in 256=r2256 = r^2. Taking the square root gives a radius of 16.

Step-by-Step Solution

1
Identify the mathematical formula for the area of a circle's sector.
Area = (θ / 360) * π * r^2
This formula relates the known area and central angle to the unknown radius.
2
Substitute the known values from the word problem into the formula.
32π = (45 / 360) * π * r^2
The scanned area is 32π and the sweep angle is 45 degrees.
3
Simplify the fraction and divide both sides by π.
32 = (1/8) * r^2
The fraction 45/360 simplifies to 1/8, and removing π from both sides simplifies the algebraic equation.
4
Multiply both sides of the equation by 8 to isolate r^2.
256 = r^2
Applying the correct inverse operation (multiplying by the reciprocal of 1/8) isolates the squared variable.
5
Take the square root of both sides to solve for r.
r = 16
Since a physical distance must be positive, the principal square root gives the correct radius.

Key Concept

Calculating the radius of a circle given the area of a sector.
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Topics

Circles

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