To find the distance between the two ships, we must convert their polar coordinates to Cartesian coordinates and then use the distance formula to find the distance between the two points.
### Step 1: Convert Polar Coordinates to Cartesian Coordinates
1. First Ship's Coordinates:
- Polar coordinates: [tex]\((8 \text{ mi}, 63^\circ)\)[/tex]
- Convert the angle from degrees to radians:
[tex]\[
\theta_1 = 63^\circ = 1.0995574287564276 \text{ radians}
\][/tex]
- Convert to Cartesian coordinates:
[tex]\[
x_1 = r_1 \cos(\theta_1) = 8 \cos(1.0995574287564276) = 3.631923997916375 \text{ mi}
\][/tex]
[tex]\[
y_1 = r_1 \sin(\theta_1) = 8 \sin(1.0995574287564276) = 7.128052193506942 \text{ mi}
\][/tex]
2. Second Ship's Coordinates:
- Polar coordinates: [tex]\((8 \text{ mi}, 123^\circ)\)[/tex]
- Convert the angle from degrees to radians:
[tex]\[
\theta_2 = 123^\circ = 2.1467549799530254 \text{ radians}
\][/tex]
- Convert to Cartesian coordinates:
[tex]\[
x_2 = r_2 \cos(\theta_2) = 8 \cos(2.1467549799530254) = -4.357112280120217 \text{ mi}
\][/tex]
[tex]\[
y_2 = r_2 \sin(\theta_2) = 8 \sin(2.1467549799530254) = 6.709364543563392 \text{ mi}
\][/tex]
### Step 2: Calculate the Distance between the Two Points
Using the distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substitute the Cartesian coordinates:
[tex]\[
d = \sqrt{(-4.357112280120217 - 3.631923997916375)^2 + (6.709364543563392 - 7.128052193506942)^2} = 8 \text{ mi}
\][/tex]
### Conclusion
After the calculations, the distance between the two ships is approximately [tex]\(8\)[/tex] miles.
Therefore, the best answer is:
A. [tex]\(\sqrt{64} \approx 8 \text{ mi}\)[/tex]
