Let's draw the scenario:
To be able to get the resultant magnitude, let's add the x and y components of the current and the boat.
Step 1: Let's break down the x and y components of the direction of the boat.
Boat: 25 miles/hr. S 65 W
x-component = 25sin(65°) = 22.66 West = -22.66 to represent that it is in the West Direction.
y-component = 25cos(65°) = 10.57 South = -10.57 to represent that it's in Downward Direction.
Current: 6 miles/hour N 15 W
x-component = 6sin(15°) = 1.55 West = -1.55 to represent that it is in the West Direction.
y-component = 6cos(15°) = 5.80 North = 5.80 to represent that it's in Upwad Direction.
Let's now add each x and y component.
Resultant x-component = -22.66 + -1.55 = -24.21 = 24.21 miles/hr. West
Resultant y-component = 5.80 - 22.66 = -16.86 = 16.86 miles/hr. South
Therefore the Resultant is equal to:
[tex]\text{ R = }\sqrt[]{24.21^2+16.86^2}[/tex][tex]R\text{ = 24.21 miles/hour}[/tex]Let's determine the angle at S-W:
[tex]\theta=\tan ^{-1}(\frac{16.86}{24.21})[/tex][tex]\theta=34.85^{\circ}[/tex]Therefore, the resultant magnitude and direction of the boat is,
[tex]24.21\text{ miles/hour at S 34.85 W}[/tex]
