As observed from the top of a lighthouse, 100 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 $\sqrt{3} = 1.732$)

Step 1: Define variables and initial setup

Let $h$ be the height of the lighthouse, which is given as 100 meters. We will use trigonometry to find the distances from the lighthouse to the ship at two different observation points.

Step 2: Calculate initial distance to the ship

When the angle of depression is $30^\circ$, let the distance of the ship from the lighthouse be $x_1$. Using the tangent function, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$, we find $x_1 = h / \tan(30^\circ)$.

Step 3: Calculate final distance to the ship

When the angle of depression changes to $45^\circ$, let the new distance of the ship from the lighthouse be $x_2$. Similarly, using the tangent function, we find $x_2 = h / \tan(45^\circ)$. Since $\tan(45^\circ) = 1$, $x_2$ is equal to $h$.

Step 4: Determine the distance travelled by the ship

The distance travelled by the ship is the difference between its initial and final distances from the lighthouse, $D = x_1 - x_2$. Substitute the values of $h$ and $\sqrt{3}$ to get the numerical answer.

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