Height of a mountain
While traveling across flat land, you notice a mountain directly in front of you. The angle of elevation to the peak is 2.5°. After you drive 18 miles closer to the mountain, the angle of elevation is 10°. Approximate the height of the mountain.
draw a diagram. if the height is h,
h/tan2.5° = h/tan10° + 18
To approximate the height of the mountain, we can use trigonometry and the concept of similar triangles. Here's how you can solve it:
Step 1: Draw a diagram
Draw a diagram representing the situation. Label the base of the mountain, the initial position where you spotted it, and the new position after driving closer. Also, mark the angle of elevation of 2.5° and 10°.
Step 2: Identify the relevant triangles
In the diagram, you have two right-angled triangles: the larger triangle representing the entire situation and the smaller triangle representing the change in position.
Step 3: Set up the ratios
Let's use the tangent function (tan) to relate the angles of elevation to the height of the mountain. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side of a right triangle.
- For the initial position: tan(theta) = height_of_mountain / distance_from_mountain
- For the new position: tan(theta+change_theta) = height_of_mountain / (distance_from_mountain - 18)
Step 4: Calculate the height of the mountain
Now, we can set up and solve these equations to find the height of the mountain.
- From the initial position: tan(2.5°) = height_of_mountain / distance_from_mountain
- From the new position: tan(10°) = height_of_mountain / (distance_from_mountain - 18)
Step 5: Apply trigonometric properties
Using trigonometric identities, we can rewrite the equations:
- For the initial position: height_of_mountain = distance_from_mountain * tan(2.5°)
- For the new position: height_of_mountain = (distance_from_mountain - 18) * tan(10°)
Step 6: Calculate the height
Substituting the known values, we can calculate the height:
- For the initial position: height_of_mountain = distance_from_mountain * tan(2.5°)
height_of_mountain = distance_from_mountain * tan(2.5°) = distance_from_mountain * 0.0436
- For the new position: height_of_mountain = (distance_from_mountain - 18) * tan(10°)
height_of_mountain = (distance_from_mountain - 18) * tan(10°) = (distance_from_mountain - 18) * 0.1763
Step 7: Approximate the height
The height of the mountain can be approximated by taking the average of the height calculated from the initial and new positions:
Approximate height_of_mountain = (height_initial + height_new) / 2
Substituting the values:
Approximate height_of_mountain = (distance_from_mountain * 0.0436 + (distance_from_mountain - 18) * 0.1763) / 2
By plugging in the value of "distance_from_mountain," you can calculate the approximate height of the mountain using the above formula.
To approximate the height of the mountain, we can use the concept of similar triangles and trigonometry.
Let's denote the height of the mountain as 'h' (in miles).
Step 1: Determine the distance between the initial observation point and when the angle of elevation is 2.5°.
The angle of elevation from the initial observation point is 2.5°. Since opposite and adjacent sides of the right triangle formed give us the tangent function, we can write:
tan(2.5°) = h / x1
Where x1 is the distance between the initial observation point and the mountain (in miles).
Step 2: Determine the distance between the new observation point (18 miles closer) and when the angle of elevation is 10°.
The angle of elevation after driving 18 miles closer to the mountain is 10°. We can apply the same logic as in Step 1:
tan(10°) = h / x2
Where x2 is the distance between the new observation point (18 miles closer) and the mountain (in miles).
Step 3: Find x1 and x2.
Since we drove 18 miles closer to the mountain, we can write:
x1 = x2 + 18
Step 4: Solve the equations.
We have two equations:
- First equation: tan(2.5°) = h / x1
- Second equation: tan(10°) = h / x2
We can substitute x1 with x2 + 18 in the first equation:
tan(2.5°) = h / (x2 + 18)
Now we have two equations with two unknowns (h and x2). We can solve for x2 using the second equation and then substitute the value of x2 in the first equation to find h.
After solving these equations, we can determine the height of the mountain.