Final answer:
To find the distance from the observer to the base of Mount Fuji, we use the tangent of the angle of elevation (30°) equals the height of the mountain (12,400 feet) divided by the distance. Solving this, the distance is approximately 21,490 feet or around 4.07 miles.
Step-by-step explanation:
The student is asking how to find the distance from the observer to the base of Mount Fuji, given the angle of elevation and the height of the mountain. This is a trigonometry problem that can be solved using the tangent function. To solve this, we need to set up an equation using the tangent of the angle of elevation, which is equal to the opposite side (height of Mount Fuji) divided by the adjacent side (distance from the base).
In this case, the height of Mount Fuji is given as 12,400 feet, and the angle of elevation is 30°. The tangent of 30° is √3/3 or approximately 0.577. Therefore, the equation will be:
tan(30°) = height / distance
0.577 = 12,400 feet / distance
Distance = 12,400 feet / 0.577 ≈ 21,490 feet
To convert the distance to miles, we divide by the number of feet in a mile (5,280 feet):
Distance ≈ 21,490 feet / 5,280 feet per mile ≈ 4.07 miles
So, the observer is approximately 4.07 miles away from the base of Mount Fuji.