Final answer:
By using the Pythagorean theorem, we can ascertain that a house located 6 miles north of the town's center and east of the cell tower is within the 3-mile coverage radius of the cell tower, as it is approximately 2.83 miles east of the tower.
Step-by-step explanation:
The cell tower is positioned 2 miles east and 5 miles north of the town's center. With a coverage radius of 3 miles, we need to determine whether a house located 6 miles north of the town's center and east of the cell tower is within the tower's coverage area. To ascertain this, we can employ the Pythagorean theorem, since the distance from the tower to the house forms a right triangle with the distances north and east of the town's center as its legs.
First, we calculate the eastward distance of the house from the cell tower. Given that the house is directly east of the cell tower and the cell tower itself is 2 miles east of the town's center, we denote the eastward distance from the cell tower to the house as 'x.' The total distance from the town's center to the house in the eastward direction would thus be '2 + x' miles.
Next, we determine the northward distance from the cell tower to the house, which is simply the difference in the northward distance of the house and the tower from the town's center, equating to '6 - 5 = 1' mile.
Using these distances, we form the equation 'x^2 + 1^2 = 3^2' to see if the house lies within or on the boundary of the cell tower's coverage. Solving for 'x,' we find that x = √(3^2 - 1^2) = √(9 - 1) = √8 which is approximately 2.83 miles. Since 'x' is smaller than the coverage radius, the house definitely lies within the coverage area of the cell tower.