To find the distance the ship is from the original starting point, we need to use the law of cosines. Here's how we can derive the expression:
When the ship starts moving due north then changes direction, it forms an angle with its previous path. Here’s the step-by-step breakdown:
1. Initial Path: The ship moves due north for [tex]\( x \)[/tex] miles.
2. Change in Direction: It then turns [tex]\( 25^\circ \)[/tex] east of north.
This means the angle between the original northward direction and the new path is [tex]\( 25^\circ \)[/tex].
3. Second Path: The ship travels [tex]\( y \)[/tex] miles in this new direction.
The law of cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(\gamma) \][/tex]
where:
- [tex]\( c \)[/tex] is the distance from the starting point to the final position,
- [tex]\( a \)[/tex] is the distance traveled along the first path,
- [tex]\( b \)[/tex] is the distance traveled along the second path,
- [tex]\( \gamma \)[/tex] is the angle between these two paths.
Here, [tex]\( a = x \)[/tex], [tex]\( b = y \)[/tex], and [tex]\( \gamma = 25^\circ \)[/tex].
Using the law of cosines, we get:
[tex]\[ c^2 = x^2 + y^2 - 2xy \cos(25^\circ) \][/tex]
Thus, the distance [tex]\( c \)[/tex] is:
[tex]\[ c = \sqrt{x^2 + y^2 - 2xy \cos(25^\circ)} \][/tex]
Therefore, the correct expression representing the distance the ship is from the original starting point is:
[tex]\[ \sqrt{x^2 + y^2 - 2xy \cos(25^\circ)} \][/tex]
By comparing this derived expression with the given choices, the correct answer is:
[tex]\[
\sqrt{x^2 + y^2 - 2xy \cos 25^{\circ}}
\][/tex]
