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.