To determine the distance between the two planes given in polar coordinates [tex]\((4 \text{ mi}, 12^{\circ})\)[/tex] and [tex]\((3 \text{ mi}, 73^{\circ})\)[/tex], follow these steps:
1. Convert Polar Coordinates to Cartesian Coordinates:\
Polar coordinates [tex]\((r, \theta)\)[/tex] can be converted to Cartesian coordinates [tex]\((x, y)\)[/tex] using the following formulas:
[tex]\[
x = r \cos(\theta)
\][/tex]
[tex]\[
y = r \sin(\theta)
\][/tex]
For the first plane:
[tex]\[
r_1 = 4 \text{ mi}, \theta_1 = 12^{\circ}
\][/tex]
Converting the angle to radians:
[tex]\[
\theta_1 = 12 \times \left(\frac{\pi}{180}\right) = \frac{12\pi}{180} \approx 0.2094 \text{ radians}
\][/tex]
Now, calculate [tex]\(x_1\)[/tex] and [tex]\(y_1\)[/tex]:
[tex]\[
x_1 = 4 \cos(0.2094) \approx 4 \times 0.9781 = 3.9124
\][/tex]
[tex]\[
y_1 = 4 \sin(0.2094) \approx 4 \times 0.2079 = 0.8316
\][/tex]
For the second plane:
[tex]\[
r_2 = 3 \text{ mi}, \theta_2 = 73^{\circ}
\][/tex]
Converting the angle to radians:
[tex]\[
\theta_2 = 73 \times \left(\frac{\pi}{180}\right) = \frac{73\pi}{180} \approx 1.2741 \text{ radians}
\][/tex]
Now, calculate [tex]\(x_2\)[/tex] and [tex]\(y_2\)[/tex]:
[tex]\[
x_2 = 3 \cos(1.2741) \approx 3 \times 0.2924 = 0.8772
\][/tex]
[tex]\[
y_2 = 3 \sin(1.2741) \approx 3 \times 0.9563 = 2.8689
\][/tex]
2. Calculate the Euclidean Distance:
The distance [tex]\(d\)[/tex] between the two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in Cartesian coordinates can be calculated using the Euclidean distance formula:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substitute [tex]\(x_1, y_1\)[/tex] and [tex]\(x_2, y_2\)[/tex] values:
[tex]\[
d = \sqrt{(0.8772 - 3.9124)^2 + (2.8689 - 0.8316)^2}
\][/tex]
[tex]\[
d = \sqrt{(-3.0352)^2 + (2.0373)^2}
\][/tex]
[tex]\[
d = \sqrt{9.2127 + 4.1516}
\][/tex]
[tex]\[
d = \sqrt{13.3643} \approx 3.6557583500674538
\][/tex]
Comparing to the provided options:
a. [tex]\(\approx 1.00 \text{ mi}\)[/tex]
b. [tex]\(\approx 61.01 \text{ mi}\)[/tex]
c. [tex]\(\approx 5.00 \text{ mi}\)[/tex]
d. [tex]\(\approx 3.66 \text{ mi}\)[/tex]
The closest match is option:
d. [tex]\(\approx 3.66 \text{ mi}\)[/tex]
Thus, the best answer is:
D
