Final answer:
Approximately 23.25 miles of the interstate highway are in the range of the cellphone tower. We use the Pythagorean theorem to calculate this by finding the length of the highway within the tower's range on one side and then doubling it.
Step-by-step explanation:
The student is asking about the area of coverage a cellphone tower provides along an interstate highway. To determine this, we can visualize the scenario as a geometry problem where we have a circle (the range of the cellphone tower) intersecting a line (the interstate highway). Since the tower is 2 miles away from the highway and has a range of 11.8 miles, we can apply the Pythagorean theorem to find the length of the highway covered within this range.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, the hypotenuse would be the radius of the circle (11.8 miles), and one of the sides would be the distance from the tower to the highway (2 miles). We can calculate the length of the other side, which will represent half the length of the highway within the range of the tower, with the following equation:
Length of highway within range = √(radius2 - distance2)
So:
Length of highway within range = √(11.82 - 22)
= √(139.24 - 4)
= √135.24
= 11.625 miles (approximately)
Since this is the length on one side of the tower, we must multiply it by 2 to get the total length of the highway covered:
Total length of highway within range = 11.625 x 2 = 23.25 miles (approximately)
Therefore, approximately 23.25 miles of the interstate highway are in the range of the cellphone tower.