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Difficulty: Very hardCircle Geometry: Arc Length and Sector Area

A goat is tethered to one of the outer corners of a flat, rectangular shed that measures 6 meters6\text{ meters} by 8 meters8\text{ meters}. The tether is 10 meters10\text{ meters} long. Assuming the goat remains outside the shed, what is the total area, in square meters, of the region the goat can graze?

  1. A
    75π75\pi
  2. B
    83π83\pi
  3. 80π80\piAnswer
  4. D
    90π90\pi
  5. E
    100π100\pi

Answer

The correct grazing area is 80π80\pi square meters.
The total grazing area is the sum of three sectors: a sector of radius 10 meters10\text{ meters} with central angle 270∘270^\circ (area 75π75\pi), a sector of radius 4 meters4\text{ meters} with central angle 90∘90^\circ (area 4π4\pi), and a sector of radius 2 meters2\text{ meters} with central angle 90∘90^\circ (area π\pi). Adding these yields 80π80\pi square meters.

Step-by-Step Solution

1
Determine the area of the main sector.
The main sector has a radius of 10 meters10\text{ meters} and a central angle of 360∘−90∘=270∘360^\circ - 90^\circ = 270^\circ. The area is 270360×π×102=75π\frac{270}{360} \times \pi \times 10^2 = 75\pi square meters.
The corner of the rectangular shed blocks 90∘90^\circ of a full circle, leaving a 270∘270^\circ sector.
2
Determine the area of the sector at the corner adjacent to the 66-meter side.
The tether wraps around the corner, leaving a remaining length of 10−6=4 meters10 - 6 = 4\text{ meters}. It sweeps through an angle of 90∘90^\circ. The area is 90360×π×42=4π\frac{90}{360} \times \pi \times 4^2 = 4\pi square meters.
When the tether extends past the adjacent corner along the 66-meter side, the pivot point becomes that corner and the tether length decreases by the side length.
3
Determine the area of the sector at the corner adjacent to the 88-meter side.
The tether wraps around the corner, leaving a remaining length of 10−8=2 meters10 - 8 = 2\text{ meters}. It sweeps through an angle of 90∘90^\circ. The area is 90360×π×22=π\frac{90}{360} \times \pi \times 2^2 = \pi square meters.
When the tether extends past the adjacent corner along the 88-meter side, the pivot point becomes that corner and the tether length decreases by the side length.
4
Check for overlap and calculate the total grazing area.
The total area is 75π+4π+π=80π75\pi + 4\pi + \pi = 80\pi square meters.
The two smaller sectors are at different corners of the rectangular shed and do not overlap. Summing the three sector areas gives the total grazing region.

Key Concept

Calculating sector areas by determining the correct radii and central angles based on geometric constraints.
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Topics

Circle Geometry: Arc Length and Sector Area

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