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This problem involves finding the distance from the boat (point \( C \)) to the shoreline. To do so, we can use the **Law of Sines** in the triangle \( \triangle ABC \), where points \( A \) and \( B \) are on the shore and point \( C \) is the boat.

### Step-by-Step Solution

1. **Identify Given Angles and Side:**
   - Angle \( \angle A = 42.3^\circ \)
   - Angle \( \angle B = 68.9^\circ \)
   - Side \( AB = 120 \) miles.

2. **Find Angle \( \angle C \):**
   Since the sum of angles in a triangle is \( 180^\circ \),
   \[
   \angle C = 180^\circ - \angle A - \angle B = 180^\circ - 42.3^\circ - 68.9^\circ = 68.8^\circ
   \]

3. **Apply the Law of Sines:**
   Using the Law of Sines, we know:
   \[
   \frac{AB}{\sin C} = \frac{AC}{\sin B} = \frac{BC}{\sin A}
   \]
   We want to find \( AC \), the distance from the boat to the shore.

   So,
   \[
   AC = \frac{AB \cdot \sin B}{\sin C}
   \]
   
4. **Plugging in the Values:**
   \[
   AC = \frac{120 \cdot \sin 68.9^\circ}{\sin 68.8^\circ}
   \]
   
5. **Calculate Using Sine Values:**
   - \( \sin 68.9^\circ \approx 0.9272 \)
   - \( \sin 68.8^\circ \approx 0.9270 \)

   Substituting these values:
   \[
   AC = \frac{120 \cdot 0.9272}{0.9270} \approx 120.03 \text{ miles}
   \]

Therefore, the distance from the boat to the shore is approximately **120 miles**.

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Would you like a deeper explanation of any part of this solution?

Here are some additional questions for further exploration:

1. How would the solution change if the distance \( AB \) was different?
2. Can you solve this problem using trigonometric identities instead of the Law of Sines?
3. What if one angle was unknown? Could we still solve it with the given side and angle?
4. How could the Law of Cosines be used to verify the distance from \( C \) to the shore?
5. If the boat moved closer or further, how would it affect \( AC \)?

**Tip:** When using the Law of Sines, ensure all angles are in the same unit (degrees or radians) to avoid calculation errors.

A boat is cruising the ocean off a straight shoreline. Points A and B are 120 mi apart on the shore, as shown. It is found that ∠A = 42.3° and ∠B = 68.9°. Find the distance from the boat to the shore.

Learn how to calculate the distance from a boat to a shoreline by applying the Law of Sines in a triangle. Given ∠A = 42.3°, ∠B = 68.9°, and AB = 120 miles, this solution covers key trigonometric concepts suitable for high school students.

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