To find the height of the lighthouse, we can solve this problem using trigonometry. Let’s denote the variables and work through the problem step by step.

Step 1: Define the Variables

• Let $h$ be the height of the lighthouse.
• The height of the man’s eye level above the water is 10 meters, so the effective height of the lighthouse above the man’s eye level is $h - 10$.
• Let the initial horizontal distance between the man and the lighthouse be $d$.

Step 2: Use the First Angle of Elevation (45°)

From the problem, the initial angle of elevation is 45°. Using the tangent of the angle in a right triangle:

$\tan(45^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h - 10}{d}$

Since $\tan(45^\circ) = 1$, we have:

$1 = \frac{h - 10}{d}$

This implies:

$d = h - 10 \quad \text{(Equation 1)}$

Step 3: Use the Second Angle of Elevation (60°)

After sailing 400 meters towards the lighthouse, the angle of elevation becomes 60°. The new horizontal distance to the lighthouse is $d - 400$. Using the tangent of the 60° angle:

$\tan(60^\circ) = \frac{h - 10}{d - 400}$

Since $\tan(60^\circ) = \sqrt{3}$, we have:

$\sqrt{3} = \frac{h - 10}{d - 400}$

This implies:

$d - 400 = \frac{h - 10}{\sqrt{3}} \quad \text{(Equation 2)}$

Step 4: Solve the System of Equations

From Equation 1, we have $d = h - 10$. Substituting this into Equation 2:

$(h - 10) - 400 = \frac{h - 10}{\sqrt{3}}$

Multiply both sides by $\sqrt{3}$ to eliminate the fraction:

$\sqrt{3} \cdot \left((h - 10) - 400\right) = h - 10$

Now expand and solve for $h$:

$\sqrt{3}(h - 10) - 400\sqrt{3} = h - 10$

Rearrange the equation:

$\sqrt{3}(h - 10) - h + 10 = 400\sqrt{3}$

Combine like terms and isolate $h$:

$h(\sqrt{3} - 1) = 10\sqrt{3} + 400\sqrt{3}$

Solve for $h$:

$h = \frac{400\sqrt{3} + 10\sqrt{3}}{\sqrt{3} - 1}$

This expression can now be simplified to find the height of the lighthouse. Let's calculate the final answer.The height of the lighthouse is approximately $970.07$ meters.

Would you like more details on the calculations or have any further questions?

Here are 5 related questions:

1. How does changing the distance to the lighthouse affect the angles of elevation?
2. What would be the new height of the lighthouse if the man’s eye level was 15 meters above the water?
3. How would the calculations differ if the ship sailed away from the lighthouse instead of towards it?
4. Can you calculate the distance between the ship and the lighthouse if the angle of elevation becomes 30°?
5. How can this problem be solved using coordinate geometry?

Tip: Remember that tangent ratios are useful for problems involving angles of elevation or depression and right triangles.