Let's solve the given problem step-by-step.

Problem Analysis:

1. A lighthouse (A) has a lookout at the top. The angle of depression from A to a boat at point C is 30°.
2. After 1 minute, the boat travels 50 m towards the lighthouse and is now at point B.
3. The angle of elevation from point B to the lighthouse lookout is 60°.

Given Data:

• Distance $BC = 50$ m.
• Angle of depression from A to C: $\angle AOC = 30^\circ$.
• Angle of elevation from B to A: $\angle ABO = 60^\circ$.

Questions:

(a) Write down the value of $x^\circ$ in the diagram.

(b) Find the height of the lighthouse.

(c) Find the speed of the boat, in meters per second, from C to B.

Solution:

(a) Value of $x^\circ$:

The value of $x^\circ$ is the angle at C, which is 60°. This is because $\angle AOC = 30^\circ$ and the angle of elevation at B to A is 60°.

Answer (a): $x = 60^\circ$.

(b) Height of the lighthouse:

Let the height of the lighthouse be $h$. From the right triangle $AOB$:

Using the tangent of the angle at B: $\tan(60^\circ) = \frac{h}{OB}.$ Since $\tan(60^\circ) = \sqrt{3}$: $OB = \frac{h}{\sqrt{3}}.$

Now, using the right triangle $AOC$:

$\tan(30^\circ) = \frac{h}{OC},$ and since $\tan(30^\circ) = \frac{1}{\sqrt{3}}$: $OC = h \cdot \sqrt{3}.$

Given that $OC = OB + BC$: $h \cdot \sqrt{3} = \frac{h}{\sqrt{3}} + 50.$

Multiplying the entire equation by $\sqrt{3}$: $3h = h + 50\sqrt{3}.$

Simplify: $2h = 50\sqrt{3},$ $h = 25\sqrt{3} \approx 25 \times 1.732 = 43.3 \text{ m}.$

Answer (b): The height of the lighthouse is approximately 43.3 m.

(c) Speed of the boat:

The boat travels from C to B, a distance of 50 m, in 1 minute (60 seconds).

Speed: $\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{50 \text{ m}}{60 \text{ s}} \approx 0.833 \text{ m/s}.$

Answer (c): The speed of the boat is approximately 0.833 m/s.

Would you like further details or explanations on any part? Here are some related questions:

1. How would the height of the lighthouse change if the angles were different?
2. Can you find the time taken if the boat was moving at a different speed?
3. What other trigonometric ratios could be used to solve similar problems?
4. How would you find the horizontal distance covered by the boat if it traveled in a different direction?
5. How do changes in elevation angles affect distance measurements in trigonometry?

Tip: Always check the units of measurement when calculating speeds, distances, or heights to ensure consistency and accuracy.