Answer:
The dog's running area is approximately 942 feet².
Step-by-step explanation:
The dog is leashed to a fixed point, the leash has a length of 20 ft, therefore he can rotate around that point at the maximum distance equal to the length of the leash. This pattern forms a circle, but there is an obstruction, which is the corner of the house. This obstruction takes an arc of the original circle, so the running area of the dog is the area of the whole circle minus the area of the arc formed by the corner of the house.
[tex]\text{dog's area} = \text{circle's area} - \text{arc's area}\\\\\text{dog's area} = pi*(20^2) - \frac{90}{360}*pi*(20^2)\\\\\text{dog's area} = \frac{270}{360}*pi*(400)\\\\\text{dog's area} = 942.477\text{ feet}^2[/tex]
The dog's running area is approximately 942 feet².
Given data
Length of leash = 20feet
The length is the same as the radius of a virtual circle the dog is bound to make
But the Corner of the house will make an angle of 90 degrees
Therefore the house will form a sector out of the virtual circle the dog can run
round
Area = πr^2
Area = 3.142*20^2
Area = 1256.8 square feet
Area of Sector Formula
The area of a sector can be calculated using the following formulas,
Area of a Sector of Circle = (θ/360º) × πr^2
where, θ is the sector angle subtended by the arc at the centre, in degrees, and 'r' is the radius of the circle.
Area of a Sector of Circle = (90/360º) × 3.142*20^2
Area of a Sector of Circle = (1/4º) × 3.142*400
Area of a Sector of Circle =3.142*100
Area of a Sector of Circle = 314.2 square feet
Hence the area the Dog can run is given as
Dog Runnable area = Area of circle - Area of Sector Subtended by the corner of the house
= 1256.8 - 314.2
= 942.6 square feet
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