A boat is sailing due east parallel to the shoreline at a speed of 11 miles per hour. At a given time, the bearing to the lighthouse is S 70° E, and 15 minutes later the bearing is S 63° E (see figure). The lighthouse is located at the shoreline. What is the distance from the boat to the shoreline? (Round your answer to one decimal place.)

To solve this problem, we can approach it using trigonometry, specifically by considering the bearings and distances as forming a right-angled triangle. Here’s the breakdown:

### Step 1: Understand the setup
- The boat is traveling due east, parallel to the shoreline, at a speed of \(11\) miles per hour.
- The bearing to the lighthouse changes from \(S 70^\circ E\) to \(S 63^\circ E\) over 15 minutes.
- The lighthouse is on the shoreline, so we need to find the distance from the boat to the shoreline.

### Step 2: Define variables
Let:
- \(d\) = the distance from the boat to the shoreline (what we need to find).
- \(x_1\) = the position of the boat at the first bearing (after 0 minutes).
- \(x_2\) = the position of the boat at the second bearing (after 15 minutes).

Since the boat is traveling at 11 miles per hour, after 15 minutes (which is \(\frac{1}{4}\) of an hour), the boat has traveled:
\[
\text{distance traveled} = 11 \times \frac{1}{4} = 2.75 \text{ miles}.
\]
Thus, the horizontal distance between the boat's two positions is 2.75 miles.

### Step 3: Set up the trigonometric relationships
At the first position, the bearing is \(S 70^\circ E\). This implies the angle between the line from the boat to the lighthouse and the horizontal (eastward) direction is \(70^\circ\).

At the second position, the bearing is \(S 63^\circ E\), meaning the angle between the line from the boat to the lighthouse and the eastward direction is \(63^\circ\).

The angles with the shoreline form a right triangle, where:
- The angle at the first position is \(70^\circ\),
- The angle at the second position is \(63^\circ\),
- The distance traveled between the two positions along the shoreline is 2.75 miles.

### Step 4: Apply the Law of Sines
We can use the Law of Sines to relate the distances and angles.

The angle between the two sightlines from the two boat positions to the lighthouse is:
\[
\Delta \theta = 70^\circ - 63^\circ = 7^\circ.
\]
Now, using the Law of Sines:
\[
\frac{2.75}{\sin(7^\circ)} = \frac{d}{\sin(63^\circ)}.
\]

### Step 5: Solve for the distance \(d\)
Now we can solve for \(d\), the distance from the boat to the shoreline:
\[
d = \frac{2.75 \times \sin(63^\circ)}{\sin(7^\circ)}.
\]
Using a calculator to find the sine values:
\[
d = \frac{2.75 \times 0.8910}{0.1219} \approx \frac{2.4503}{0.1219} \approx 20.1 \text{ miles}.
\]

Thus, the distance from the boat to the shoreline is approximately **20.1 miles**.

Would you like further details or clarifications?

### Follow-up Questions:
1. How does the Law of Sines work in solving triangles?
2. How do bearings and angles of depression work in navigation problems?
3. How do we find sine and cosine values using a calculator?
4. Why do we subtract angles when calculating the difference in bearings?
5. Can the Law of Cosines be used in this situation?

### Tip:
When working with bearings, always draw a clear diagram to visualize the problem better!

A boat is sailing due east parallel to the shoreline at a speed of 11 miles per hour. At a given time, the bearing to the lighthouse is S 70° E, and 15 minutes later the bearing is S 63° E. The lighthouse is located at the shoreline. What is the distance from the boat to the shoreline? (Round your answer to one decimal place.)

Solve a navigation problem involving a boat sailing east at 11 mph. Learn how to calculate the distance from the boat to the shoreline using trigonometry and the Law of Sines. The problem covers bearings and involves a 15-minute change in angle from S 70° E to S 63° E.

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