Two boats are 70m and on opposite sides of a lighthouse.from the two boats,the angles of elevation of the lighthouse are 71 and 45 respectively.find the height of the lighthouse.
draw a diagram. It shows that if the height is h, then
h cot71° + h cot45° = 70
To find the height of the lighthouse, we can use the concept of trigonometry. Let's assume that the height of the lighthouse is 'h' meters.
First, let's consider the boat that forms an angle of elevation of 71 degrees with the lighthouse. We can imagine a right-angled triangle formed between the boat, the lighthouse, and a point on the ground directly below the lighthouse.
In this triangle, the side opposite the angle of elevation is the height of the lighthouse 'h' and the side adjacent to the angle is the distance between the boat and the lighthouse. Let's call this distance 'x'.
Using the tangent function, we have:
tan(71) = h / x
Next, let's consider the other boat that forms an angle of elevation of 45 degrees with the lighthouse. In a similar way, we can imagine another right-angled triangle formed between this boat, the lighthouse, and the same point on the ground.
In this triangle, the side opposite the angle of elevation is again the height of the lighthouse 'h'. The side adjacent to the angle is the distance between the other boat and the lighthouse, which is also 'x'.
Using the tangent function, we have:
tan(45) = h / x
We now have two equations:
1) tan(71) = h / x
2) tan(45) = h / x
To solve for 'h', we can divide equation (1) by equation (2):
(tan(71) / tan(45)) = (h / x) / (h / x)
Since tan(45) = 1 (opposite side and adjacent side are equal in a 45-45-90 triangle),
we have:
tan(71) = h / h
tan(71) = 1
h = tan(71)
Now you can use a scientific calculator or an online calculator to find the tangent of 71 degrees. The result will give you the height of the lighthouse.