We can approach this problem by breaking down the two displacements of the ship into their respective x- and y-components and then adding them together to find the net displacement.
For the first displacement, the ship traveled 25° South of West for 250 miles. This can be broken down into an x-component and a y-component as follows:
x = 250 cos(25°) (to the west) y = -250 sin(25°) (to the south)
For the second displacement, the ship changed direction to 70° East of South and traveled 45 miles further. This can also be broken down into an x-component and a y-component:
x = 45 cos(70°) (to the east) y = -45 sin(70°) (to the south)
To find the net displacement, we can add the x-components and y-components separately:
total x = 250 cos(25°) + 45 cos(70°) total y = -250 sin(25°) - 45 sin(70°)
We can use these values to find the distance of the ship from its starting point by using the Pythagorean theorem:
distance = sqrt((total x)^2 + (total y)^2)
Substituting the values from above and evaluating:
distance = sqrt((250 cos(25°) + 45 cos(70°))^2 + (-250 sin(25°) - 45 sin(70°))^2)
distance ≈ 272.8 miles
To find the direction of the ship from its starting point, we can use the inverse tangent function to find the angle:
angle = atan(total y / total x)
Substituting the values from above and evaluating:
angle ≈ -65.1°
Since the angle is negative, we know that the direction is to the west of south. Therefore, the ship is approximately 272.8 miles away from its starting point in a direction that is 65.1° west of south.