Final answer:
To find the height of the mountain, we can use trigonometry to set up equations based on the given information. By solving these equations, we can determine that the height of the mountain is approximately 2 km.
Step-by-step explanation:
To find the height of the mountain, we can use trigonometry. Let's assume the height of the mountain is h km. From the information given, we have:
Step 1: At the foot of the mountain, the elevation of its summit is 45 degrees.
Step 2: After ascending 2 km towards the mountain up a slope of 30 degrees inclination, the elevation is found to be 60 degrees.
Using the trigonometric ratios, we can set up the following equations:
Tan 45 = h / x, where x is the distance from the foot of the mountain to the point where the elevation is 60 degrees.
Tan 30 = h / (x + 2), where (x + 2) is the total distance traveled from the foot of the mountain.
Simplifying the equations, we have:
1 = h / x, and
√3 = h / (x + 2).
Dividing the second equation by the first equation, we get:
√3 = (x + 2) / x.
By cross multiplication, we have:
√3x = x + 2.
√3x - x = 2.
x (√3 - 1) = 2.
x = 2 / (√3 - 1).
Substituting this value of x into the first equation, we can find the height of the mountain:
h = x.
Simplifying, we have:
h ≈ 2 km.
Therefore, the height of the mountain is approximately 2 km.