Final answer:
The distance between the two lookout towers is calculated using the Pythagorean theorem based on their positions relative to the fire, resulting in a distance of 7.28 km, which does not match the provided multiple-choice options.
Step-by-step explanation:
The question involves finding the distance between two points on a coordinate system, which corresponds to a typical problem in mathematics, specifically coordinate geometry or the Pythagorean theorem. To determine how far apart the two lookout towers are, we need to analyze their positions relative to the fire. The first tower is located 5 km west and 3 km north of the fire, while the second tower is 3 km west and 4 km south of the fire. To find the distance between the two towers, we can consider this a right-angled triangle problem where the towers' positions with respect to the fire give us the lengths of two sides of the triangle.
The difference in the westward distance between the two towers is the absolute value of the difference between their westward distances from the fire, which is |5 km - 3 km| = 2 km. The difference in the northward distance is the sum of their northward and southward distances from the fire, which is 3 km + 4 km = 7 km. The distance between the two towers is the hypotenuse of the right-angled triangle formed by these sides, which can be calculated using the Pythagorean theorem: √(2² + 7²) = √(4 + 49) = √53 = 7.28 km.
However, this calculated distance of 7.28 km is not one of the options provided in the question, which means there might be a typo in the question, or possibly a mistake in the calculation. The closest answer to the calculated value is 7.6 km, which could be a rounding of the exact answer. Without the options aligning with the calculated distance, I cannot confidently indicate which of the options a), b), c), or d) is correct.