let the height of the cliff be h feet
to the top of the lighthouse
tan 50.2 = (h+40)/75
h+ 40 = 75tan 50.2
h = 75tan 50.2 - 40
= ........
to the top of the lighthouse
tan 50.2 = (h+40)/75
h+ 40 = 75tan 50.2
h = 75tan 50.2 - 40
= ........
Let's define the following:
h = height of the cliff
d = distance from the boat to the base of the cliff
The angle of elevation to the top of the lighthouse from the boat is 50.2 degrees. Therefore, the angle of depression from the top of the lighthouse to the boat is also 50.2 degrees.
Now, we can set up a right triangle with the given information. The vertical side of the triangle represents the height of the lighthouse (40 feet), the horizontal side represents the distance from the boat to the base of the cliff (75 feet + d), and the hypotenuse represents the line of sight.
Since we have two angles and one side of the triangle, we can use the tangent function to find the height of the cliff:
tan(50.2 degrees) = h / (75 + d)
We want to solve for h, so let's isolate it:
h = (75 + d) * tan(50.2 degrees)
Now we can substitute the given values into the equation and solve for h:
h = (75 + d) * tan(50.2 degrees)
h = (75 + 75) * tan(50.2 degrees)
h = 150 * tan(50.2 degrees)
h ≈ 150 * 1.3152
h ≈ 197.28
The height of the cliff is approximately 197.28 feet.
Let's define our triangle:
The boat represents the bottom vertex of the triangle.
The top of the lighthouse represents the top vertex of the triangle.
The base of the cliff represents the other vertex of the triangle.
Given:
Angle of elevation: 50.2 degrees
Distance from boat to base of the cliff: 75 feet
We can use the tangent function, which relates the angle of elevation to the opposite and adjacent sides of the triangle:
tan(angle) = opposite / adjacent
In this case, the height of the cliff is the opposite side and the distance from the boat to the base of the cliff is the adjacent side.
So, we have:
tan(50.2) = height of cliff / 75
To find the height of the cliff, we rearrange the equation:
height of cliff = tan(50.2) * 75
Now we can calculate the height of the cliff. Let's plug in the values:
height of cliff = tan(50.2) * 75
height of cliff ≈ 1.331 * 75
height of cliff ≈ 99.825 feet
Therefore, the height of the cliff is approximately 99.825 feet.