Answer:
Rounding to the nearest whole degree, the rescue boat should travel at an angle of approximately 41 degrees off of due east to take the shortest route to the ship.
Explanation:
We can use trigonometry to solve this problem. Let's draw a diagram to represent the situation:
A (rescue boat)
| \
| \
| \ C (disabled ship)
24 mi | \
| \
| \
|θ \
B-----------D (Port Lincoln)
48 mi
In the diagram, point B represents Port Lincoln, point C represents the disabled ship, and point A represents the rescue boat. Point D is the intersection of the eastward path and the northward path taken by the ship.
We are given that BD = 24 miles and CD = 48 miles. We want to find the angle θ, which is the angle between the line segments AB and AD.
To find θ, we can use the law of cosines:
cos(θ) = (BD² + CD² - AD²) / (2 x BD x CD)
Substituting the given values, we get:
cos(θ) = (24² + 48² - AD²) / (2 x 24 x 48)
Simplifying, we get:
cos(θ) = 0.75
To solve for θ, we can take the inverse cosine of both sides:
θ = cos⁻¹(0.75)
Using a calculator, we get:
θ ≈ 41.41°
Rounding to the nearest whole degree, the rescue boat should travel at an angle of approximately 41 degrees off of due east to take the shortest route to the ship.