Question: Two sailboats leave a harbor at the same time. The first sails 20 mph in a direction of…

Two sailboats leave a harbor at the same time. The first sails 20 mph in a direction of \(328^\circ\). The second sails at 26 mph in a direction of \(203^\circ\). Assuming that both boats maintain their current speed and heading, after 2 hours, how far apart are the boats? Round to the nearest whole mile.

Solution

First, calculate the distance each sailboat travels in 2 hours. \[ \text{Distance}_1 = 20 \, \text{mph} \times 2 \, \text{hours} = 40 \, \text{miles} & \] \[ \text{Distance}_2 = 26 \, \text{mph} \times 2 \, \text{hours} = 52 \, \text{miles} & \] Next, determine the components of each sailboat’s displacement. For the first sailboat traveling at 328°, the components are: \[ x_1 = 40 \cos(328°) & \] \[ y_1 = 40 \sin(328°) & \] For the second sailboat traveling at 203°, the components are: \[ x_2 = 52 \cos(203°) & \] \[ y_2 = 52 \sin(203°) & \] Calculate the difference in the components to find the displacement between the two sailboats. \[ \Delta x = x_1 - x_2 & \] \[ \Delta y = y_1 - y_2 & \] Finally, use the Pythagorean theorem to find the distance between the two sailboats. \[ \text{Distance} = \sqrt{(\Delta x)^2 + (\Delta y)^2} & \] After performing the calculations, the distance between the two sailboats after 2 hours is approximately 69 miles.