Choose one of the following trig application problems. 5. A ferry is used to transport guests from the dock to two hotels across a large lake. The hotels are located 550 m apart from each other.The first hotel is at a 49 degree angle between the dock and the second hotel. The second hotel is at a 56 degree angle between the dock and the first hotel. How far is each hotel from the dock? ( 6 marks)

## Step 1: Visualizing the Problem<br />### We have a triangle formed by the dock and the two hotels. Let D be the dock, H1 be the first hotel, and H2 be the second hotel. We are given the distance H1H2 = 550m, angle DH1H2 = 49°, and angle DH2H1 = 56°. We need to find DH1 and DH2.<br /><br />## Step 2: Finding the remaining angle<br />### The sum of angles in a triangle is 180°. Therefore, angle H1DH2 = 180° - (49° + 56°) = 180° - 105° = 75°.<br /><br />## Step 3: Applying the Sine Rule<br />### The sine rule states that $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. In our case, we have:<br />$\frac{DH1}{\sin 56°} = \frac{DH2}{\sin 49°} = \frac{550}{\sin 75°}$<br /><br />## Step 4: Calculating DH1<br />### $DH1 = \frac{550 \cdot \sin 56°}{\sin 75°} = \frac{550 \cdot 0.829}{0.966} \approx 472.58$m<br /><br />## Step 5: Calculating DH2<br />### $DH2 = \frac{550 \cdot \sin 49°}{\sin 75°} = \frac{550 \cdot 0.755}{0.966} \approx 428.67$m