The bearing from A to C is S 58​° E. The bearing from A to B is N 86​° E. The bearing from B to C is S 32​° W. A plane flying at 250 mph takes 3.1 hr to go from A to B. Find the distance from A to C

To solve the problem, we can use the law of sines in a triangle formed by the points \( A \), \( B \), and \( C \). The given information will allow us to find the distance \( AC \).

### Step 1: Determine the angles in the triangle
- **Bearing from \( A \) to \( B \)**: \( N 86^\circ E \). This means the angle with respect to the North direction is \( 86^\circ \).
- **Bearing from \( A \) to \( C \)**: \( S 58^\circ E \). This means the angle with respect to the South direction is \( 58^\circ \).
- **Bearing from \( B \) to \( C \)**: \( S 32^\circ W \). This means the angle with respect to the South direction is \( 32^\circ \).

### Step 2: Calculate the angles of triangle \( ABC \)
- **Angle \( \angle BAC \)**: The angle between the bearings from \( A \) to \( B \) and from \( A \) to \( C \) can be found by subtracting \( 86^\circ \) from \( 180^\circ - 58^\circ \).
  
  \[
  \angle BAC = 180^\circ - (86^\circ + 58^\circ) = 36^\circ
  \]

- **Angle \( \angle ABC \)**: The angle between the bearings from \( A \) to \( B \) and from \( B \) to \( C \) can be found by subtracting \( 86^\circ \) from \( 180^\circ - 32^\circ \).

  \[
  \angle ABC = 180^\circ - (86^\circ + 32^\circ) = 62^\circ
  \]

- **Angle \( \angle ACB \)**: This is the remaining angle in the triangle and can be calculated as:
  
  \[
  \angle ACB = 180^\circ - (36^\circ + 62^\circ) = 82^\circ
  \]

### Step 3: Calculate the distance \( AB \)
The distance \( AB \) can be calculated using the speed and time of the plane.

\[
AB = \text{Speed} \times \text{Time} = 250 \text{ mph} \times 3.1 \text{ hours} = 775 \text{ miles}
\]

### Step 4: Use the Law of Sines to find \( AC \)
The Law of Sines states:

\[
\frac{AC}{\sin(\angle ABC)} = \frac{AB}{\sin(\angle ACB)}
\]

Substituting the known values:

\[
\frac{AC}{\sin(62^\circ)} = \frac{775}{\sin(82^\circ)}
\]

Solve for \( AC \):

\[
AC = \frac{775 \times \sin(62^\circ)}{\sin(82^\circ)}
\]

### Step 5: Calculate the value of \( AC \)
Now, we can calculate the value of \( AC \):

\[
AC \approx \frac{775 \times 0.8829}{0.9903} \approx 690.8 \text{ miles}
\]

### Final Answer:
The distance from \( A \) to \( C \) is approximately **690.8 miles**.

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Would you like more details on any part of this process or do you have any further questions?

### Related Questions:
1. How would the result change if the plane's speed was different?
2. What if the angle \( \angle ABC \) was measured differently?
3. Can you derive the same result using vectors instead of trigonometry?
4. How would you calculate the bearing from \( C \) to \( A \)?
5. What if there was a wind affecting the plane's speed and direction?

**Tip:** When solving bearing problems, always visualize the angles on a diagram to avoid confusion between different directional references.

The bearing from A to C is S 58° E. The bearing from A to B is N 86° E. The bearing from B to C is S 32° W. A plane flying at 250 mph takes 3.1 hr to go from A to B. Find the distance from A to C.

Learn how to calculate the distance between two points using bearings and the Law of Sines. This example problem involves finding the distance from point A to point C, given bearings and speed information. Suitable for high school students in Grades 10-12, this detailed solution covers key concepts in trigonometry and navigation.

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