The bearing from A to C is S 62​° E. The bearing from A to B is N 86​° E. The bearing from B to C is S 28​° W. A plane flying at 250 mph takes 2.8 hr to go from A to B. Find the distance from A to C.
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Part 1
The distance from A to C is 
  
enter your response here mi.
​(Round to the nearest integer as​ needed.)

Let's first break down the information and solve the problem step-by-step:

### Given:
- **Bearing from A to C:** \( S 62^\circ E \) (This means an angle of \( 62^\circ \) east of south)
- **Bearing from A to B:** \( N 86^\circ E \) (This means an angle of \( 86^\circ \) east of north)
- **Bearing from B to C:** \( S 28^\circ W \) (This means an angle of \( 28^\circ \) west of south)
- **Speed of the plane:** 250 mph
- **Time taken from A to B:** 2.8 hours

### Step 1: Calculate the distance from A to B
The formula for distance is:
\[
\text{Distance} = \text{Speed} \times \text{Time}
\]
\[
\text{Distance from A to B} = 250 \times 2.8 = 700 \text{ miles}
\]

### Step 2: Construct a triangle
We now have a triangle formed by points A, B, and C. To find the distance from A to C, we can apply the Law of Cosines. However, first we need to find the angle between the bearings at point A (between \( A \to C \) and \( A \to B \)).

- **Angle between A to C and A to B:**
  The bearing from A to C is \( S 62^\circ E \), and the bearing from A to B is \( N 86^\circ E \).
  The angle between these two bearings is:
  \[
  \theta = 180^\circ - (86^\circ + 62^\circ) = 180^\circ - 148^\circ = 32^\circ
  \]

### Step 3: Use the Law of Cosines
The Law of Cosines states:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)
\]
Where:
- \( a \) and \( b \) are the lengths of two sides of the triangle (in this case, both \( a \) and \( b \) are the distance from A to B, which is 700 miles),
- \( \theta \) is the angle between them (which we found to be \( 32^\circ \)),
- \( c \) is the distance from A to C (which we are trying to find).

Now applying the values:
\[
c^2 = 700^2 + 700^2 - 2 \cdot 700 \cdot 700 \cdot \cos(32^\circ)
\]

Let's calculate this.The distance from A to C is approximately 386 miles (rounded to the nearest integer).

Do you want more details or have any questions? Here are five related questions to explore further:

1. How does the Law of Cosines work in non-right-angled triangles?
2. Can the Law of Sines be used for this problem? Why or why not?
3. How do bearings relate to angular measurements in navigation?
4. What other methods could be used to solve distance problems in navigation?
5. What are common mistakes when interpreting bearings in trigonometry problems?

**Tip:** Always make sure to properly interpret bearings as angles from either the north or south line before calculating the included angle.

The bearing from A to C is S 62° E. The bearing from A to B is N 86° E. The bearing from B to C is S 28° W. A plane flying at 250 mph takes 2.8 hr to go from A to B. Find the distance from A to C.

Solve a real-world trigonometry problem involving bearings and distance calculation. Learn how to use the Law of Cosines to find the distance from point A to point C, given bearings from A to B, A to C, and B to C. This problem is suitable for high school students in Grades 10-12.

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