Question

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

ships,
a. 64 8 mi
C. 11.31 mi
b. 3600.00 mi
d. 4.14 mi
Please select the best answer

Answer 1

A.
`√(64)=8` miles

Explanation:

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

`d = √(((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:

`√(((r_1 cos\theta_1-r_2 cos \theta_2)^2+(r_1 sin \theta_1-r_2 sin \theta_2)^2)\\=√((r_1^2+r_2^2-2r_1r_2(cos \theta_1 cos \theta_2+sin\theta_1 sin \theta_2) )\\= √(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=√(8^2+8^2-2(8)(8)cos (63 -123))\\=√(128-128cos (-60))\\=√(64)=8 mile`

The closest option is A.
`√(64)=8` miles