A dog is tied to a leash that is hooked to the outside corner of a barn that measures 12 ft.x20 ft. The length of the leash is 16 ft. What is the maximum area in which the dog can wander?
see 1st link below, this same problem was illustrated yesterday.
To find the maximum area in which the dog can wander, we need to determine the shape and size of this area.
Since the leash is tied to the outside corner of the barn and cannot pass through walls, the shape of the area will be a quarter circle.
The radius of the quarter circle is equal to the length of the leash, which is 16 ft.
We can use the formula for the area of a circle to find the maximum area:
Area = (π x r^2) / 4
Where π is approximately 3.14159 and r is the radius.
Plugging in the values:
Area = (3.14159 x 16^2) / 4
= (3.14159 x 256) / 4
= 804.24 / 4
= 201.06 square feet
Therefore, the maximum area in which the dog can wander is approximately 201.06 square feet.
To find the maximum area in which the dog can wander, we need to determine the shape that represents the boundary of the wandering area. In this case, the dog is tied to a leash that is hooked to the outside corner of a barn. As the dog wanders, the leash will create an arc shape.
To determine the maximum area, we first need to find the radius of the arc created by the leash. This can be found using the Pythagorean theorem. The leash forms the hypotenuse of a right triangle, with the sides of the barn acting as the other two sides.
Using the Pythagorean theorem, we can find the length of the diagonal of the barn:
diagonal^2 = 12ft^2 + 20ft^2
diagonal^2 = 144ft^2 + 400ft^2
diagonal^2 = 544ft^2
diagonal = √544ft
diagonal ≈ 23.32ft
Since the leash has a length of 16ft, the radius of the arc created by the leash is half of that length:
radius = 16ft / 2
radius = 8ft
Now that we have the radius, we can calculate the area of the maximum wandering area, which is the area of the circular sector created by the leash. The formula for the area of a sector is given by:
area = (θ/360°) * π * r^2
Where θ is the central angle of the sector, in degrees, and r is the radius of the circle.
In this case, since the leash is 16ft and the radius is 8ft, the central angle can be found using the equation:
sin(θ/2) = (8ft / 16ft)
θ/2 = sin^(-1)(1/2)
θ/2 = 30°
Therefore, the central angle θ is 60°. We can now calculate the maximum area:
area = (60°/360°) * π * (8ft)^2
area = (1/6) * π * 64ft^2
area ≈ 33.51ft^2
Hence, the maximum area in which the dog can wander is approximately 33.51 square feet.