Final answer:
The total area that the dog can reach outside of the building is approximately 128.66 square meters.
Step-by-step explanation:
To find the total area that the dog can reach, we need to determine the shape of the region. Since the dog is tied to the midpoint of one of the long sides of the building, it can move in a circular path with a radius of 10 meters.
This forms a semicircle. The total area of the region that the dog can reach is the sum of the area of the semicircle and the area of the rectangle, minus the overlapping area.
The area of the semicircle can be calculated using the formula A = 1/2 * π * r^2, where r is the radius of the semicircle. In this case, r = 10 meters, so the area of the semicircle is 1/2 * π * 10^2 = 50π square meters.
The area of the rectangle is given as 4 meters by 6 meters, so the area is 4 * 6 = 24 square meters.
The overlapping area is the area of the rectangle that falls within the semicircle.
Since the rectangle is longer than the diameter of the semicircle (6 meters vs. 10 meters), the overlapping area is determined by the area of the semicircle with a width of 6 meters. This can be calculated as 1/2 * π * (6/2)^2 = 9π square meters.
Therefore, the total area that the dog can reach is 50π + 24 - 9π = 41π square meters, or approximately 128.66 square meters.