Question

Solve each problem. See Examples 1 and 2. Distance between Two Ships A ship leaves its home port and sails on a bearing of S 61°50'. Another ship leaves the same port at the same time and sails on a bearing of N 28°10'E. If the first ship sails at 24.0 mph and the second sails at 28.0 mph, find the distance between the two ships after 4 hr.

Accepted Answer

Convert the bearings into standard angles measured counterclockwise from the positive x-axis (East). For the first ship with bearing S 61°50', this corresponds to an angle of 180° + 61°50' (since it's measured clockwise from North to South), so calculate the exact angle in degrees. For the second ship with bearing N 28°10' E, this corresponds to an angle of 90° - 28°10' (since it's measured clockwise from North to East), so calculate this angle as well. Calculate the distance each ship has traveled after 4 hours by multiplying their speeds by time: Distance = Speed × Time. For the first ship, multiply 24.0 mph by 4 hours; for the second ship, multiply 28.0 mph by 4 hours. Express the position of each ship in Cartesian coordinates (x, y) using their distances and angles. Use the formulas: $$x = r 	imes \cos(	heta)$$ and $$y = r 	imes \sin(	heta)$$, where $$r is $$the distance traveled and $$	heta is $$the angle in radians (convert degrees to radians). Find the difference in the x-coordinates and y-coordinates between the two ships to get the components of the vector representing the distance between them: $$\Delta x = x_2$ - x_1$ and $$\Delta y = $$y_2$$ - $$y_1. $$Use the Pythagorean theorem to find the distance between the two ships: $$	ext{Distance} = \sqrt{(\Delta x)^2$ + (\Delta y)^2$}. $$This will give the straight-line distance between the ships after 4 hours.

Risposta video verificata per un problema simile:

Concetti chiave

Bearing and Direction in Navigation

Vector Representation of Displacement

Law of Cosines for Distance Calculation