3. A ship has sailed 520 mi from the port in Jacksonville. From the ship, the captain records an angle measurement between the port in Jacksonville and the port at a private island as $$ 52^{\circ} $$. He knows the angle from the port in Jacksonville between the ship and the island port is $$ 78^{\circ} $$. a. Make and label a diagram to model this scenario. b. To the nearest mile, how far is the ship from the island port? Explain.

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### Problem 3

A ship has sailed 520 miles from the port in Jacksonville. The captain records an angle measurement between the port in Jacksonville and the port at a private island as $$ 52^{\circ} $$. He also knows the angle from the port in Jacksonville between the ship and the island port is $$ 78^{\circ} $$.

---

#### **a. Diagram**

To model this scenario, we can use a triangle where:

- **J** represents the **Port in Jacksonville**.
- **S** represents the **Ship**.
- **P** represents the **Port at the Private Island**.

**Steps to Draw the Diagram:**

1. **Draw Triangle JSP**:
   - **Point J** (Jacksonville Port)
   - **Point S** (Ship) is 520 miles from **J**.
   - **Point P** (Private Island Port).

2. **Label the Sides**:
   - **JS** (distance from Jacksonville to Ship) = **520 miles**.
   - **SP** (distance from Ship to Private Island) = *Unknown* (This is what we need to find).
   - **JP** (distance from Jacksonville to Private Island) = *Can be derived if needed*.

3. **Label the Angles**:
   - **$$\angle J$$** (at Jacksonville Port) = $$78^{\circ}$$ (angle between Ship and Private Island as viewed from Jacksonville).
   - **$$\angle S$$** (at Ship) = $$52^{\circ}$$ (angle between Jacksonville Port and Private Island as viewed from the Ship).
   - **$$\angle P$$** (at Private Island Port) = $$180^{\circ} - 78^{\circ} - 52^{\circ} = 50^{\circ}$$.

**Visual Representation:**

```
          P
         / \
        /   \
   78° /     \ 50°
      /       \
     J---------S
        520 mi
         52°
```

- **J**: Jacksonville Port
- **S**: Ship
- **P**: Private Island Port
- **JS** = 520 miles
- **$$\angle J = 78^{\circ}$$**
- **$$\angle S = 52^{\circ}$$**
- **$$\angle P = 50^{\circ}$$**

---

#### **b. Calculating the Distance from the Ship to the Island Port**

We need to find the distance **SP** from the ship to the private island port. We'll use the **Law of Sines**, which relates the sides and angles of a triangle:

$$
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
$$

Where:
- $$ a, b, c $$ are the sides opposite angles $$ A, B, C $$ respectively.

**Applying the Law of Sines to Triangle JSP:**

1. **Identify the Corresponding Sides and Angles:**
   - Side **JS** = 520 miles is opposite **$$\angle P = 50^{\circ}$$**.
   - Side **SP** is opposite **$$\angle J = 78^{\circ}$$**.

2. **Set Up the Proportion:**
$$
\frac{SP}{\sin(78^{\circ})} = \frac{520}{\sin(50^{\circ})}
$$

3. **Solve for SP:**
$$
SP = \frac{520 \times \sin(78^{\circ})}{\sin(50^{\circ})}
$$

4. **Calculate the Sines:**
   - $$\sin(78^{\circ}) \approx 0.9781$$
   - $$\sin(50^{\circ}) \approx 0.7660$$

5. **Plug in the Values:**
$$
SP = \frac{520 \times 0.9781}{0.7660} \approx \frac{507.752}{0.7660} \approx 662.6 \text{ miles}
$$

6. **Round to the Nearest Mile:**
$$
SP \approx 663 \text{ miles}
$$

**Answer:** To the nearest mile, the ship is **663 miles** from the island port.
The ship is approximately 663 miles from the island port.

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