To find the distance $BC$ between the birds at points $B$ and $C$, we can first determine the coordinates of points $B$ and $C$ using their respective bearings and distances from point $A$.

Step 1: Determine Coordinates of B and C

1. Point B:

   • Distance: $10$ km
   • Bearing: $N80^\circ E$ (which means $80^\circ$ clockwise from North)
   • Coordinates:
     • $x_B = 10 \sin(80^\circ)$
     • $y_B = 10 \cos(80^\circ)$

2. Point C:

   • Distance: $9$ km
   • Bearing: $S70^\circ W$ (which means $70^\circ$ clockwise from South, or $180^\circ + 70^\circ = 250^\circ$ from North)
   • Coordinates:
     • $x_C = 9 \sin(250^\circ)$
     • $y_C = 9 \cos(250^\circ)$

Step 2: Calculate Coordinates

Now let's calculate the coordinates for points $B$ and $C$:

• For point $B$:

  • $x_B = 10 \sin(80^\circ) \approx 10 \times 0.9848 \approx 9.848$ km
  • $y_B = 10 \cos(80^\circ) \approx 10 \times 0.1736 \approx 1.736$ km
  • Thus, $B \approx (9.848, 1.736)$.

• For point $C$:

  • $x_C = 9 \sin(250^\circ) \approx 9 \times (-0.9397) \approx -8.457$ km
  • $y_C = 9 \cos(250^\circ) \approx 9 \times (-0.3420) \approx -3.078$ km
  • Thus, $C \approx (-8.457, -3.078)$.

Step 3: Find Distance BC

Using the distance formula: $BC = \sqrt{(x_B - x_C)^2 + (y_B - y_C)^2}$

Substituting the coordinates: $BC = \sqrt{(9.848 - (-8.457))^2 + (1.736 - (-3.078))^2}$ $= \sqrt{(9.848 + 8.457)^2 + (1.736 + 3.078)^2}$ $= \sqrt{(18.305)^2 + (4.814)^2}$ $= \sqrt{334.522 + 23.167}$ $= \sqrt{357.689} \approx 18.9 \text{ km}$

Conclusion

The distance $BC$ between the two birds is approximately $18.9$ km.

Would you like more details on any part of the calculation? Here are some related questions:

1. How would the distance change if the bearings were altered?
2. What are the implications of the bearing directions on the birds' paths?
3. How do you convert bearings to standard coordinate angles?
4. What other methods can be used to find distances between points on a plane?
5. Can you explain how to visualize this problem using a diagram?

Tip: Always ensure your angles are in the correct mode (degrees vs. radians) when using a calculator!