The angles of elevation to the top of a mountain are measured from beacons A(22°) and B(40°) at sea. If the beacons are 125m apart, how high is the mountain?
Assuming B is between A and the mountain, then the height h is found from
h cot22° - h cot40° = 125
To find the height of the mountain, we can use trigonometry. Let's denote the height of the mountain as h.
First, let's consider the triangle formed by beacon A, the top of the mountain, and a point C on the mountain directly below beacon A. This is a right-angled triangle.
Using the angle of elevation from beacon A, we can write:
tan(22°) = h / x
where x is the distance from beacon A to the point C on the mountain.
Similarly, considering the triangle formed by beacon B, the top of the mountain, and a point D on the mountain directly below beacon B, we have:
tan(40°) = h / (125 - x)
where 125 - x represents the distance from beacon B to the point D on the mountain.
Since the height of the mountain h is the same in both triangles, we can equate the two expressions for h:
h / x = h / (125 - x)
To solve for x, we can cross-multiply and simplify the equation:
h * (125 - x) = h * x
125h - hx = hx
125h = 2hx
Now, we can solve for x:
x = 125h / (2h)
x = 62.5
So the distance from beacon A to the point C on the mountain is 62.5 meters.
Now, we can use this value of x to find the height of the mountain. Using the equation from the first triangle:
tan(22°) = h / x
tan(22°) = h / 62.5
Solving for h:
h = 62.5 * tan(22°)
h ≈ 24.07 meters
Therefore, the height of the mountain is approximately 24.07 meters.
To find the height of the mountain, we can make use of the angles of elevation and the distance between the beacons. Let's break down the problem step by step:
1. Draw a diagram: Draw a rough diagram to visualize the problem. Label points A and B as the two beacons at sea level. Label point C as the top of the mountain. Also, mark the distance between the beacons as 125m.
A C B
|______|______|
125m
2. Define the variables: Let's assign variables to the unknown quantities. Let h be the height of the mountain.
3. Look for a right triangle: Notice that triangle ABC forms a right triangle, with angle CAB (angle of elevation at beacon A) and angle CBA (angle of elevation at beacon B).
4. Apply trigonometry: We can use the tangent function to relate the ratio of the height to the horizontal distance between the beacons.
In triangle ABC, for beacon A:
tan(angle CAB) = height / distance AB
In triangle ABC, for beacon B:
tan(angle CBA) = height / distance AB
Since angle CAB = 22° and angle CBA = 40°, we have two equations:
tan(22°) = h / 125m
tan(40°) = h / 125m
5. Solve the equations: We can solve the equations simultaneously to find the value of h. Let's solve using the first equation:
tan(22°) = h / 125m
Multiply both sides by 125m:
125m * tan(22°) = h
Use a calculator to find the value of tan(22°) and calculate h.
6. Calculate the height: Substitute the calculated value of h into either of the original equations and solve for h.
tan(40°) = h / 125m
Multiply both sides by 125m:
125m * tan(40°) = h
Use a calculator to find the value of tan(40°) and calculate h.
7. The height of the mountain: The value of h obtained from either equation will be the height of the mountain.
Remember to round your final answer to an appropriate number of significant figures based on the given values and keep track of units.