Question 19

Explanation

Geometric Visualization

Position Analysis

Let:

• $A$: Top of the hill
• $B$: Base of the hill (perpendicular to the ground)
• $C$: Initial boat position
• $D$: Pier position (final boat position)

Given:

• Boat travel distance $CD = 2\text{ km}$.
• Initial line of sight distance $AC = 12\text{ km}$.
• Initial elevation angle at $C$ is $37^\circ$.
• Final elevation angle at $D$ is $53^\circ$.

We need to find the distance from the base of the hill to the end of the pier, which is the length $BD$.

Distance Calculation

Consider the right-angled triangle $ABC$. We can use the cosine ratio for angle $C$ ($37^\circ$):

The length $BC$ is the total distance from the base of the hill to the initial boat position, consisting of the distance from the pier to the hill base ($BD$) and the boat's travel distance ($CD$).

Substitute the known values into the cosine equation:

Thus, the distance from the base of the hill to the end of the pier is $7.6\text{ km}$.