A bridge is 70m long. From the ends of the bridge the angles of depression of a point on a river under the bridge are 41 degrees and 48 degrees. How high is the bridge above the river?
The point is obviously not under the MIDDLE of the bridge because the angles are not equal. I will assume however that it is directly under the bridge.
Let x be the distace by which the point is displaced from being below the middle of the bridge. Let h be the height of the bridge.
h/(35 + x) = sin 41
h/(35 - x) = sin 48
(35-x)/(35+x) = sin 41/sin48 = 0.88281
35 - x = 30.899 + .88281x
Solve for x and then solve for h.
hlkh
To find the height of the bridge above the river, we can use trigonometry and the concept of angles of depression.
Let's first label the given information:
- Length of the bridge: 70m.
- Angle of depression from one end of the bridge: 41 degrees.
- Angle of depression from the other end of the bridge: 48 degrees.
To solve this problem, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the bridge, and the adjacent side is half of the length of the bridge.
Let's denote the height of the bridge as h. From one end of the bridge, using the angle of depression of 41 degrees, we can write the following equation:
tan(41) = h / (70/2)
Simplifying the equation:
tan(41) = h / 35
Now, from the other end of the bridge, using the angle of depression of 48 degrees, we can write another equation:
tan(48) = h / (70/2)
Simplifying the equation:
tan(48) = h / 35
We have two equations, both expressing h/35. Therefore, we can equate these two equations:
tan(41) = tan(48)
Now we can solve for h by multiplying both sides by 35:
h = tan(41) * 35
Using a calculator, we can find that tan(41) ≈ 0.869 and rounding our answer:
h ≈ 0.869 * 35
h ≈ 30.4
Thus, the height of the bridge above the river is approximately 30.4 meters.