Final answer:
The location of Balloon C relative to the tower is approximately 2 miles east and 2 miles north. The correct answer is option a).
Step-by-step explanation:
To find the location of Balloon C relative to the tower, we first need to find the coordinates of Balloon A and Balloon B.
Given that Balloon A is 5 miles east and 8 miles south of the tower, its coordinates are (5, -8). Balloon B is 7 miles west and 10 miles north of the tower, so its coordinates are (-7, 10).
Next, we can find the distance between Balloon A and Balloon B:
Distance = sqrt((5 - (-7))^2 + (-8 - 10)^2)
= sqrt(144 + 324)
= sqrt(468)
= 6sqrt(13) miles.
Since Balloon C is located four-ninths the distance from Balloon A to Balloon B, the distance between Balloon A and Balloon C is
(4/9)(6sqrt(13))
= 8/3sqrt(13) miles.
To find the coordinates of Balloon C, we can use similar triangles. The vertical distance between Balloon A and Balloon C is 8/9 times the vertical distance between Balloon A and Balloon B, which is -8/3 miles. Similarly, the horizontal distance between Balloon A and Balloon C is 8/9 times the horizontal distance between Balloon A and Balloon B, which is -36/9 miles.
Therefore, the location of Balloon C relative to the tower is approximately 2 miles east and 2 miles north.