We can solve this problem by breaking it into two parts: the distance and direction (bearing) of each leg of the flight, and then combining them using trigonometry.

Part A: Distance Calculation

1. First leg (A to B):

   • The plane flies 2.6 hours at 110 mph on a bearing of N70E.
   • Distance $d_1$ traveled in the first leg is: $d_1 = 110 \, \text{mph} \times 2.6 \, \text{hours} = 286 \, \text{miles}$
   • Since the bearing is N70E, the direction forms an angle of 70° from the north (or 20° from the east). The components of this distance in the x (eastward) and y (northward) directions are: $x_1 = d_1 \times \cos(70^\circ) = 286 \times \cos(70^\circ)$ $y_1 = d_1 \times \sin(70^\circ) = 286 \times \sin(70^\circ)$

2. Second leg (B to C):

   • The plane flies 2.8 hours at 110 mph on a bearing of S20E.
   • Distance $d_2$ traveled in the second leg is: $d_2 = 110 \, \text{mph} \times 2.8 \, \text{hours} = 308 \, \text{miles}$
   • The bearing S20E is 20° east of south, meaning it forms a 20° angle with the south (or 70° from the east). The components of this distance in the x (eastward) and y (southward) directions are: $x_2 = d_2 \times \cos(20^\circ) = 308 \times \cos(20^\circ)$ $y_2 = -d_2 \times \sin(20^\circ) = -308 \times \sin(20^\circ) \, \text{(since it's southward)}$

3. Total displacement: The total displacement of the plane from the starting point is found by summing the x and y components: $x_{\text{total}} = x_1 + x_2$ $y_{\text{total}} = y_1 + y_2$ The straight-line distance (using the Pythagorean theorem) is: $D = \sqrt{x_{\text{total}}^2 + y_{\text{total}}^2}$

Part B: Bearing Calculation

The bearing of the plane from the original location can be found by calculating the angle of the resultant displacement vector from the north: $\theta = \arctan \left( \frac{x_{\text{total}}}{y_{\text{total}}} \right)$ If the final x-component is positive and y-component is negative, the bearing will be measured clockwise from north, which will give us the bearing SθE.

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Let me calculate these values for you.### Part A: Distance Calculation The total distance the plane is from its starting point after both legs of the flight is approximately 420.309 miles.

Part B: Bearing Calculation

The bearing of the plane from its original location (measured clockwise from due north) is approximately 67.121°.

Let me know if you'd like more details or have any questions!

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Here are five related questions to expand your understanding:

1. How would the total distance change if the plane flew at a higher speed?
2. What happens to the bearing if the second leg's direction was S70E instead of S20E?
3. How would the total distance differ if the plane flew a shorter second leg (e.g., 2 hours instead of 2.8)?
4. Can we determine how long the entire journey took in minutes?
5. What would be the x and y components if the plane flew westward instead of eastward in the first leg?

Tip: When calculating bearings, always ensure you're measuring the angle from due north or south, adjusting for east/west directions accordingly.