Questions: A ship is moving in a straight line toward the Point Cove lighthouse. The measure of the angle of elevation from the bridge of the ship to the lighthouse beacon is 25°. Later, from a point 600 ft closer, the angle of elevation is 47°. What is the m< BOT?

Solution Steps

First, we calculate the tangent values for the angles of elevation from the ship to the lighthouse. The tangent of the angles is given by: \[ \tan(25^{\circ}) \approx 0.4663 \] \[ \tan(47^{\circ}) \approx 1.0724 \]

Let \( d \) be the initial distance from the ship to the lighthouse and \( h \) be the height of the lighthouse. From the first observation point, we have: \[ h = d \cdot \tan(25^{\circ}) \] From the second observation point, which is 600 ft closer, we have: \[ h = (d - 600) \cdot \tan(47^{\circ}) \]

We can set the two equations for \( h \) equal to each other: \[ d \cdot \tan(25^{\circ}) = (d - 600) \cdot \tan(47^{\circ}) \] Solving this system of equations yields: \[ d \approx 1061.6442 \text{ ft} \] \[ h \approx 495.0528 \text{ ft} \]

To find the angle \( \text{BOT} \), we use the relationship: \[ \tan(\text{BOT}) = \frac{h}{d} \] Substituting the values of \( h \) and \( d \): \[ \tan(\text{BOT}) = \frac{495.0528}{1061.6442} \] Calculating the angle gives: \[ \text{BOT} \approx 25^{\circ} \]

Final Answer