To find the speed of the boat, we can use trigonometric relationships. The problem involves changing angles of elevation, which means we can use the tangent function, as it relates the height of the lighthouse to the horizontal distance of the boat from the base of the lighthouse.

Step 1: Set up the initial conditions

Let:

• $h = 206.6$ m (height of the lighthouse)
• $d_1$ = initial distance of the boat from the lighthouse when the angle of elevation is $60^\circ$
• $d_2$ = distance of the boat from the lighthouse when the angle of elevation is $45^\circ$
• $v$ = speed of the boat (which we need to find)
• $t = 120$ seconds (time taken for the angle to change)

Step 2: Apply trigonometry

For the angle $60^\circ$: $\tan(60^\circ) = \frac{h}{d_1}$ $\sqrt{3} = \frac{206.6}{d_1}$ $d_1 = \frac{206.6}{\sqrt{3}} \approx 119.3 \text{ meters}$

For the angle $45^\circ$: $\tan(45^\circ) = \frac{h}{d_2}$ $1 = \frac{206.6}{d_2}$ $d_2 = 206.6 \text{ meters}$

Step 3: Calculate the distance traveled by the boat

The distance traveled by the boat $\Delta d$ is the difference between $d_2$ and $d_1$: $\Delta d = d_2 - d_1 = 206.6 - 119.3 = 87.3 \text{ meters}$

Step 4: Calculate the speed of the boat

The speed $v$ of the boat is given by the formula: $v = \frac{\Delta d}{t} = \frac{87.3 \text{ meters}}{120 \text{ seconds}} \approx 0.7275 \text{ meters per second}$

Thus, the speed of the boat is approximately 0.7275 meters per second.

Would you like more details or have any questions?

Here are 8 related questions:

1. How does the tangent function relate to angles of elevation in real-world problems?
2. What is the significance of the change in angle from 60° to 45° in this problem?
3. How would the speed change if the angle of elevation changed more quickly?
4. What assumptions are made in this problem to simplify the calculations?
5. How would the solution differ if the height of the lighthouse were different?
6. Can the method used here be applied to objects moving in three dimensions?
7. How would you find the speed if the angle of elevation changed from 45° to 30° instead?
8. What other types of real-world problems can be solved using trigonometry?

Tip: Always verify the units of measurement in problems involving speed, distance, and time to ensure consistency in your calculations.