While sailing, Donna sees a lighthouse and calculates that the angle of elevation to the lighthouse is 3d, as shown in the accompanying diagram. When she sails her boat 700 feet closer to the lighthouse, she finds that the angle
To answer this question, we need to use trigonometric ratios, specifically tangent. The tangent of an angle helps us relate the angle of elevation to the distance between the observer and the object.
Let's call the distance between Donna's boat and the lighthouse, initially, as "x" feet.
We know the angle of elevation, which is 3d. The tangent of an angle is defined as the opposite side divided by the adjacent side.
In this case, the opposite side is the vertical distance between Donna's eye level and the top of the lighthouse. We can call this distance "y" feet.
The adjacent side is the horizontal distance between Donna's boat and the lighthouse, which we called "x" feet.
So, we have the equation: tangent(3d) = y / x.
Now, let's consider what happens when Donna sails her boat 700 feet closer to the lighthouse. This means the new distance between her boat and the lighthouse is (x - 700) feet.
At this new position, she observes a different angle of elevation, let's call it "e".
Using the same reasoning as before, we can write the equation: tangent(e) = y / (x - 700).
Now, we have two equations:
1. tangent(3d) = y / x
2. tangent(e) = y / (x - 700)
We can solve these equations to find the value of "d" and "e".
To find the value of "d", we rearrange the first equation:
y = x * tangent(3d)
Substituting this value of "y" into the second equation, we get:
tangent(e) = (x * tangent(3d)) / (x - 700)
Now, we can substitute values for "e" and solve for "d".
To find "e", we need more information or values. Unfortunately, the information provided in the question is incomplete, as the angle "e" is missing.