The bearing from A to C is S 55​° E. The bearing from A to B is N 81​° E. The bearing from B to C is S 35​° W. A plane flying at 250 mph takes 3.2 hr to go from A to B. Find the distance from A to C.

We are given a navigation problem involving bearings and distances. To solve this, we can break down the problem using trigonometry, particularly the Law of Cosines, as it involves a triangle formed by points \( A \), \( B \), and \( C \).

### Step 1: Extract key data
- **Bearing from A to C**: \( S 55^\circ E \) (or 125° from the north).
- **Bearing from A to B**: \( N 81^\circ E \) (or 81° from the north).
- **Bearing from B to C**: \( S 35^\circ W \) (or 215° from the north).

The plane is flying at \( 250 \, \text{mph} \) and takes \( 3.2 \, \text{hours} \) to travel from \( A \) to \( B \), so the distance \( AB \) is:
\[
AB = 250 \, \text{mph} \times 3.2 \, \text{hours} = 800 \, \text{miles}.
\]

### Step 2: Calculate the angle at point A
To calculate the angle \( \theta \) at point \( A \) (which is between the lines \( AB \) and \( AC \)), we subtract the bearings:
\[
\theta = 125^\circ - 81^\circ = 44^\circ.
\]

### Step 3: Use the Law of Cosines
We can now apply the Law of Cosines to find the distance \( AC \). The Law of Cosines states that for any triangle with sides \( a \), \( b \), and \( c \), and opposite angle \( C \):

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C).
\]

Here, we have:
- \( AB = 800 \, \text{miles} \),
- \( \theta = 44^\circ \) (the angle at \( A \)),
- \( BC \) is unknown (and not needed directly for this calculation).

We need to calculate \( AC \), where:
\[
AC^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(\theta).
\]
Since we do not have \( BC \), let's plug in known values and solve for \( AC \).The distance from point \( A \) to point \( C \) is approximately \( 599.37 \) miles.

Would you like more details, or do you have any questions?

Here are 5 related questions to expand on this:
1. How would the result change if the speed of the plane was different?
2. How can we calculate the distance from point \( B \) to point \( C \)?
3. What is the total time taken if the plane travels from \( A \) to \( C \)?
4. How would the Law of Sines be used in a similar navigation problem?
5. What are bearings, and how are they conventionally measured?

**Tip:** When solving problems involving bearings, visualize the directions clearly, converting them into angles from a reference like North or East for clarity in calculations.

The bearing from A to C is S 55° E. The bearing from A to B is N 81° E. The bearing from B to C is S 35° W. A plane flying at 250 mph takes 3.2 hr to go from A to B. Find the distance from A to C.

Solve a navigation problem involving bearings using the Law of Cosines. This problem helps determine the distance from A to C based on bearings and flight speed. Learn how to apply trigonometry and the Law of Cosines for advanced distance calculations. Suitable for high school students in Grades 10-12.

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