Question

The locations, given in polar coordinates, for two ships are (8 mi, 639) and (8 mi, 1239). Find the distance between the two

Answer 1

Answer:

A. $\sqrt{64}=8$ miles

Step-by-step explanation:

Given two Cartesian coordinates $(x_1,y_1)\&(x_2,y_2)$, the distance between the points is given as:

$d = \sqrt{((x_1-x_2)^2+(y_1-y_2)^2)}$

Converting to polar coordinates

$(x_1,y_1) = (r_1 cos \theta_1, r_1 sin \theta_1)\\(x_2,y_2) = (r_2 cos \theta_2, r_2 sin \theta_2)$

Substitution into the distance formula gives:

$\sqrt{((r_1 cos\theta_1-r_2 cos \theta_2)^2+(r_1 sin \theta_1-r_2 sin \theta_2)^2}\\=\sqrt{(r_1^2+r_2^2-2r_1r_2(cos \theta_1 cos \theta_2+sin\theta_1 sin \theta_2) }\\= \sqrt{r_1^2+r_2^2-2r_1r_2cos (\theta_1 -\theta_2)}$

In the given problem,

$(r_1,\theta_1)=(8 mi, 63^0) \:and\: (r_2,\theta_2)=(8 mi, 123^0 ).$

$Distance=\sqrt{8^2+8^2-2(8)(8)cos (63 -123)}\\=\sqrt{128-128cos (-60)}\\=\sqrt{64}=8 mile$

The closest option is  A. $\sqrt{64}=8$ miles