Final answer:
The total distance the hiker has to travel to reach the summit is approximately 8,000 ft.
Explanation:
In order to find the total distance, we need to use the Pythagorean Theorem, which 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 other two sides.
In this case, the vertical ascent of 4,000 ft is one of the sides, and the distance traveled along the incline is the other side. Therefore, we can set up the equation: a² + b² = c², where a = 4,000 ft and c is the total distance traveled. We can solve for b by using the angle of incline, a = 120 degrees, and the fact that the sum of the angles in a triangle is 180 degrees. This means that the angle opposite to the 4,000 ft side is 60 degrees (180 - 120 = 60). We can then use trigonometric functions to find the value of b: b = 4,000 ft * tan(60) = 4,000 ft * √3 ≈ 6,928.2 ft. Now, we can plug this value into our original equation: 4,000² + 6,928.2² = c². After solving for c, we get c ≈ 8,000 ft.
Therefore, the total distance the hiker has to travel to reach the summit is approximately 8,000 ft.