The bearing from A to C is S 57​° E. The bearing from A to B is N 88​° E. The bearing from B to C is S 33​° W. A plane flying at 250 mph takes 2.8 hr to go from A to B. Find the distance from A to C.

The distance from A to C is

enter your response here mi.

To solve this problem, we will use trigonometry, specifically the law of sines, since we are dealing with a triangle and we have bearings that will allow us to find the angles of this triangle.

### Step 1: Determine the angles of the triangle.
- The bearing from A to C is \( \text{S } 57^\circ \text{ E} \). This means the angle between the south direction and the line AC is \( 57^\circ \).
- The bearing from A to B is \( \text{N } 88^\circ \text{ E} \). This means the angle between the north direction and the line AB is \( 88^\circ \).
- The bearing from B to C is \( \text{S } 33^\circ \text{ W} \). This means the angle between the south direction and the line BC is \( 33^\circ \).

### Step 2: Calculate the angle at A.
- The angle \( \angle BAC \) can be found by subtracting the bearing from A to B from the bearing from A to C.
  
  \[
  \angle BAC = 88^\circ - 57^\circ = 31^\circ
  \]

### Step 3: Calculate the angle at B.
- The angle \( \angle ABC \) is calculated by subtracting the bearing from B to C from the bearing from A to B.
  
  \[
  \angle ABC = 88^\circ + 33^\circ = 121^\circ
  \]

### Step 4: Calculate the angle at C.
- The angle \( \angle ACB \) in the triangle can be calculated as:
  
  \[
  \angle ACB = 180^\circ - \angle BAC - \angle ABC
  \]
  \[
  \angle ACB = 180^\circ - 31^\circ - 121^\circ = 28^\circ
  \]

### Step 5: Calculate the distance from A to B.
- The distance from A to B can be calculated using the speed and time:
  
  \[
  \text{Distance from A to B} = \text{Speed} \times \text{Time} = 250 \text{ mph} \times 2.8 \text{ hr} = 700 \text{ miles}
  \]

### Step 6: Apply the law of sines to find the distance from A to C.
- The law of sines states:
  
  \[
  \frac{\text{AB}}{\sin(\angle ACB)} = \frac{\text{AC}}{\sin(\angle ABC)}
  \]
  
  Solving for \( \text{AC} \):
  \[
  \text{AC} = \text{AB} \times \frac{\sin(\angle ABC)}{\sin(\angle ACB)}
  \]
  
  Substituting the values:
  \[
  \text{AC} = 700 \times \frac{\sin(121^\circ)}{\sin(28^\circ)}
  \]

### Step 7: Calculate the distance using a calculator.
Let's compute this.

\[
\sin(121^\circ) \approx 0.8572
\]
\[
\sin(28^\circ) \approx 0.4695
\]
\[
\text{AC} = 700 \times \frac{0.8572}{0.4695} \approx 700 \times 1.826 \approx 1278.2 \text{ miles}
\]

So, the distance from A to C is approximately **1278.2 miles**.

Would you like more details or have any questions?

### Related Questions
1. How would the distance change if the speed of the plane was different?
2. What would be the bearing from C to A?
3. How can the Law of Cosines be used in this situation?
4. If the plane had taken a different route, what other factors would need consideration?
5. What is the significance of knowing the bearings in navigation?

### Tip
Always double-check your angle calculations when working with bearings, as small errors can lead to significant discrepancies in the final results.

Learn how to calculate the distance from point A to point C using bearings and trigonometry. Given bearings from A to C, A to B, and B to C, and the speed of a plane from A to B, find the distance from A to C using the law of sines and trigonometric calculations.

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