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Difficulty: MediumCircles

A drone is tethered to a fixed landing pad and can fly within a maximum circular boundary with an area of 400π400\pi square meters. If a safety protocol requires the tether to be shortened by 88 meters, what is the maximum area of the new circular flight boundary?

  1. 144π square meters144\pi \text{ square meters}Answer
  2. B
    256π square meters256\pi \text{ square meters}
  3. C
    336π square meters336\pi \text{ square meters}
  4. D
    24π square meters24\pi \text{ square meters}

Answer

The new flight boundary has an area of 144π square meters144\pi \text{ square meters}.
The initial area of 400π400\pi square meters indicates the original tether length (radius) was 2020 meters, because πr2=400π\pi r^2 = 400\pi. Shortening the tether by 88 meters results in a new radius of 1212 meters. The new area is calculated by squaring the new radius and multiplying by π\pi, yielding π(12)2=144π\pi(12)^2 = 144\pi square meters.

Step-by-Step Solution

1
Find the original radius using the area formula A=πr2A = \pi r^2.
400π=πr2400\pi = \pi r^2, so r2=400r^2 = 400, which means r=20r = 20 meters.
The original tether length represents the radius of the circular boundary.
2
Calculate the new radius by subtracting 88 meters.
20−8=1220 - 8 = 12 meters.
The safety protocol shortens the tether, directly reducing the flight radius.
3
Calculate the area of the new circular boundary.
A=π(12)2=144π square metersA = \pi(12)^2 = 144\pi \text{ square meters}.
The problem asks for the maximum area of the new boundary.

Key Concept

Area of a circle and its relationship to the radius.
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Topics

Circles

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