Martha observed a boat from the top of a 74.5 m cliff. She noted that the angle of depression to the boat was 22.70How far is the boat from the base of the cliff ?Round to the nearest 2 decimal places.
Question
Martha observed a boat from the top of a 74.5 m cliff. She noted that the angle of depression to the boat was 22.70How far is the boat from the base of the cliff ?Round to the nearest 2 decimal places.
Solution
To solve this problem, we can use the tangent of the angle of depression, which is the ratio of the opposite side (the height of the cliff) to the adjacent side (the distance from the boat to the base of the cliff).
The formula for tangent is:
tan(θ) = opposite/adjacent
We know the angle of depression (θ) is 22.7 degrees and the height of the cliff (opposite side) is 74.5 m. We want to find the distance from the boat to the base of the cliff (adjacent side), which we'll call x.
So, we can set up the equation as follows:
tan(22.7) = 74.5/x
To solve for x, we rearrange the equation:
x = 74.5/tan(22.7)
Now, we just need to plug in the values and calculate:
x = 74.5/tan(22.7) = 185.97 m
So, the boat is approximately 185.97 meters from the base of the cliff, rounded to the nearest two decimal places.
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