As observed from the top of a lighthouse, $100 \text{ m}$ above sea level, the angle of depression of a ship, sailing directly towards it, changes from $30^\circ$ to $45^\circ$. Determine the distance travelled by the ship during the period of observation. (Use $\sqrt{3} = 1.732$) (angle of depression, trigonometry, right triangle, distance, lighthouse)

Step 1: Define variables and set up the initial triangle

Let $AB$ be the height of the lighthouse, which is given as $100 \text{ m}$. Let $C$ be the initial position of the ship and $D$ be its final position. The angles of depression are given as $30^\circ$ and $45^\circ$. Due to alternate interior angles, the angle of elevation from the ship to the top of the lighthouse will be equal to the angle of depression. So, $\angle ADB = 30^\circ$ and $\angle ACB = 45^\circ$.

Step 2: Calculate distance BC using $\triangle ABC$

In the right-angled triangle $ABC$, we can use the tangent function to relate the height of the lighthouse to the distance $BC$. Since $\tan(45^\circ) = 1$, we find that the distance $BC$ is equal to the height of the lighthouse, $100 \text{ m}$.

Step 3: Calculate distance BD using $\triangle ABD$

Similarly, in the right-angled triangle $ABD$, we use the tangent function. We know that $\tan(30^\circ) = \frac{1}{\sqrt{3}}$. This allows us to calculate the total distance $BD$ from the lighthouse to the ship's initial position.

Step 4: Calculate the distance travelled by the ship

The distance travelled by the ship is the difference between its initial distance from the lighthouse ($BD$) and its final distance from the lighthouse ($BC$). We subtract $BC$ from $BD$ to find $CD$.

Step 5: Substitute the value of $\sqrt{3}$ and calculate the final distance

Finally, we substitute the given value of $\sqrt{3} = 1.732$ into the expression for $CD$ and perform the arithmetic to get the numerical value of the distance travelled by the ship.

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