Answer: The mountain is approximately 384.21 feet high.
We can solve this problem using trigonometry and the tangent function!
Step 1: Identify what we know.
Matt's angle of elevation = 35 degrees
Susie's angle of elevation = 42 degrees
Susie is 122 feet closer to the mountain than Matt.
Let:
h = Height of the mountain
x = Horizontal distance from Susie to the mountain
Since Matt is 122 feet farther away, his distance is x + 122.
Step 2: Set up two equations using tangent.
Remember:
Tangent = Opposite / Adjacent
For Susie:
tan(42) = h / x
Therefore:
h = x tan(42)
For Matt:
tan(35) = h / (x + 122)
Therefore:
h = (x + 122) tan(35)
Step 3: Set the equations equal to each other.
Since both equations represent the same mountain height:
x tan(42) = (x + 122) tan(35)
Using a calculator in degree mode:
0.9004x = 0.7002(x + 122)
Distribute:
0.9004x = 0.7002x + 85.43
Subtract 0.7002x from both sides:
0.2002x = 85.43
Solve for x:
x ≈ 426.70 feet
Step 4: Calculate the mountain's height.
Now substitute the distance into Susie's equation:
h = x tan(42)
h ≈ 426.70 x 0.9004
h ≈ 384.21 feet
Final Answer: The mountain is approximately 384.21 feet tall!
Why does this work?
Notice that Susie has the larger angle of elevation (42 degrees) because she is closer to the mountain. Matt has the smaller angle (35 degrees) because he is farther away.
By using both angles and the 122-foot difference in their positions, we can calculate the height without actually climbing the mountain!
This assumes both people are on level ground, looking toward the same point directly below the peak, and that the mountain's height is measured vertically from that ground level.
