The bearing from A to C is S 63​° E. The bearing from A to B is N 88​° E. The bearing from B to C is S 27​° W. A plane flying at 250 mph takes 3.3 hr to go from A to B. Find the distance from A to C.


To solve for the distance from A to C, we can use the law of cosines. Here's how we can approach the problem:

### Step 1: Breakdown of the Problem
- We are given the bearings:
  - From A to C: \( S 63^\circ E \)
  - From A to B: \( N 88^\circ E \)
  - From B to C: \( S 27^\circ W \)
- The plane's speed is 250 mph, and it takes 3.3 hours to travel from A to B.

### Step 2: Calculate the Distance from A to B
The speed of the plane is 250 mph, and the time it takes to travel from A to B is 3.3 hours. We can calculate the distance from A to B:

\[
AB = 250 \, \text{mph} \times 3.3 \, \text{hours} = 825 \, \text{miles}
\]

### Step 3: Calculate the Angle at Point A (∠CAB)
- The bearing from A to C is \( S 63^\circ E \), which translates to a direction 63° clockwise from south.
- The bearing from A to B is \( N 88^\circ E \), which is almost directly east.

To find the angle between the two directions (∠CAB), we calculate the difference between the two bearings:
\[
\text{Angle} \, \angle CAB = 63^\circ + (90^\circ - 88^\circ) = 63^\circ + 2^\circ = 65^\circ
\]

### Step 4: Apply the Law of Cosines
We can now use the law of cosines to find the distance from A to C. The law of cosines states:

\[
AC^2 = AB^2 + BC^2 - 2(AB)(BC) \cdot \cos(\angle CAB)
\]

But we first need the distance \( BC \), which requires a bit more information about the relative angles. To proceed more accurately, additional information about the triangle's other side lengths or angles might be needed.

Would you like me to continue with assumptions or do you have more details on the distances or angles?

The bearing from A to C is S 63° E. The bearing from A to B is N 88° E. The bearing from B to C is S 27° W. A plane flying at 250 mph takes 3.3 hr to go from A to B. Find the distance from A to C.

This problem involves calculating the distance between two points using bearings and the law of cosines. Given the bearings from A to C, A to B, and B to C, and the speed of a plane, we use trigonometry and bearings concepts to find the distance from A to C. Suitable for students in Grades 11-12.

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