A man standing on top of a mountain 1500m high observes the angle of depression of the top of a steeple to be 40 degrees, if the height of the steeple is 50m, how far is it from the mountain ?
difference in height = 1450
tan 40 = 1450/x
To find the distance between the mountain and the steeple, we can use trigonometry. Let's label the unknown distance as "x".
Based on the given information:
- The man standing on top of the mountain observes the angle of depression of the top of the steeple to be 40 degrees.
- The height of the steeple is 50m.
- The mountain has a height of 1500m.
Now, we can set up a trigonometric equation using the tangent function:
tan(40°) = opposite / adjacent.
In this case, the opposite side is the height of the steeple (50m) and the adjacent side is the unknown distance (x).
tan(40°) = 50 / x.
We can now solve for x by rearranging the equation:
x = 50 / tan(40°).
Using a calculator, we can find the value of tan(40°) ≈ 0.8391.
x = 50 / 0.8391 ≈ 59.59.
Therefore, the distance between the mountain and the steeple is approximately 59.59 meters.
To find the distance from the mountain to the steeple, we can use trigonometry and the concept of an angle of depression.
Let's denote the distance from the man to the steeple as "x" (the unknown that we are trying to find).
Since the man is standing on top of a mountain, we can consider a right-angled triangle formed by the man, the top of the steeple, and a point directly below the man on the ground. The height of the mountain can be considered as the height of this right-angled triangle.
The angle of depression (40 degrees) is the angle between the line of sight from the man to the top of the steeple and a horizontal line drawn from the man's eye level.
Now, we have two sides of the triangle: the height of the mountain (1500m) and the height of the steeple (50m). We need to find the third side, which is the distance (x) from the man to the steeple.
To calculate this, we can use the tangent ratio, which states that the tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle.
In our case, the tangent of the angle of depression (40 degrees) is equal to the ratio of the height of the steeple (50m) to the distance from the man to the steeple (x):
tan(40 degrees) = 50 / x
To find x, we can rearrange the equation:
x = 50 / tan(40 degrees)
Now we can calculate x using a scientific calculator or an online calculator:
x ≈ 50 / tan(40 degrees) ≈ 50 / 0.8391 ≈ 59.652
Therefore, the distance from the mountain to the steeple is approximately 59.652 meters.