In travelling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is 4.5°. After you drive 13 miles closer to the mountain, the angle of elevation is 9.5°.
X1 = Distance from closest point to bottom of mountain.
X1+13 = Distance from farthest point to bottom of mountain.
Tan9.5 = h/X1, h = X1*Tan9.5.
Tan4.5 = h/(X1+13), (X1+13)*Tan4.5.
h = X1*Tan9.5 = (X1+13)*Tan4.5
X1 = 0.507(X1+13) = 0.507X1+6.59.
X1-0.501X1 = 6.59.
0.5X1 = 6.59.
X1 = 13.2 Mi.
Tan9.5 = h/13.2
h = 13.2*Tan9.5 = 2.21 Miles.
To solve this problem, we can use trigonometry, specifically the concepts of angle of elevation and right triangles.
First, let's define some variables:
- Let "x" represent the distance between your original position and the base of the mountain.
- Let "h" represent the height of the mountain.
We are given two pieces of information:
1. When you are at your original position, the angle of elevation to the peak of the mountain is 4.5°.
2. After driving 13 miles closer to the mountain, the angle of elevation to the peak is 9.5°.
Using these pieces of information, we can set up two right triangles.
In the first scenario (when you are at your original position):
- The angle of elevation is 4.5°.
- The opposite side is "h" (height of the mountain).
- The adjacent side is "x" (distance from your position to the base of the mountain).
In the second scenario (after driving 13 miles closer to the mountain):
- The angle of elevation is 9.5°.
- The opposite side remains "h" (height of the mountain).
- The adjacent side is now "x + 13" (distance from your new position to the base of the mountain).
We can use the tangent function, which relates the angle of elevation to the opposite and adjacent sides of a right triangle:
- tan(angle of elevation) = opposite/adjacent
Using the tangent function, we can set up two equations:
For the first scenario:
tan(4.5°) = h / x
For the second scenario:
tan(9.5°) = h / (x + 13)
Now, we have a system of two equations with two variables (h and x).
To find the height of the mountain (h), we can solve this system of equations. We can rearrange both equations to solve for h:
For the first equation:
h = x * tan(4.5°)
For the second equation:
h = (x + 13) * tan(9.5°)
Since both equations equal h, we can set them equal to each other:
x * tan(4.5°) = (x + 13) * tan(9.5°)
Solve this equation for x to find the distance from your original position to the base of the mountain. Then, substitute the value of x into either of the equations to find the height of the mountain (h).