A coast guard patrol boat is 14.8 km east of the Brier Island Lighthouse. A disabled yacht is 7.5km south of the lighthouse.
a) How far is the patrol boat from the yacht, to the nearest tenth of a kilometer?
b) At what angle south of due west, to the nearest degree, should the patrol boat travel to reach the yacht?
a. d^2 = (14.8)^2 + (7.5)^2 = 275.19
d = sqrt(275.19) = 16.6 km.
b. Tan B = 7.5/14.8 = 0.5068, B = 26.9
deg.
To find the distance between the patrol boat and the yacht, we can use the Pythagorean theorem since we have a right triangle formed by the patrol boat, the yacht, and the lighthouse. The distance between the patrol boat and the yacht is the hypotenuse of this triangle.
a) To find the distance, we can use the formula:
distance = sqrt((horizontal distance)^2 + (vertical distance)^2)
We know that the horizontal distance is 14.8 km (east) and the vertical distance is 7.5 km (south).
distance = sqrt((14.8 km)^2 + (7.5 km)^2)
distance = sqrt(219.04 km^2 + 56.25 km^2)
distance = sqrt(275.29 km^2)
distance ≈ 16.6 km (to the nearest tenth of a kilometer)
Therefore, the patrol boat is approximately 16.6 km from the yacht.
b) To find the angle at which the patrol boat should travel to reach the yacht, we can use trigonometry. Since the patrol boat needs to travel south of due west, we are interested in finding the angle south of due west.
We can use the tangent function to find this angle:
tan(angle) = (vertical distance) / (horizontal distance)
Plugging in the values we know:
tan(angle) = 7.5 km / 14.8 km
angle = arctan(7.5 km / 14.8 km) (inverse tangent)
angle ≈ 27 degrees (to the nearest degree)
Therefore, the patrol boat should travel at an angle of approximately 27 degrees south of due west to reach the yacht.
To solve this problem, we can use the Pythagorean theorem to find the distance between the patrol boat and the yacht. Then, we can use trigonometry to find the angle at which the boat should travel. Let's go step by step.
a) To find the distance between the patrol boat and the yacht, we can use the Pythagorean theorem. The distance is equal to the square root of the sum of the squares of the two sides.
So, the distance between the patrol boat and the yacht is:
Distance = sqrt((14.8 km)^2 + (7.5 km)^2)
Distance = sqrt(219.04 km^2 + 56.25 km^2)
Distance = sqrt(275.29 km^2)
Distance ≈ 16.59 km
Therefore, the patrol boat is approximately 16.59 km away from the yacht.
b) To find the angle, we can use inverse trigonometric functions. In this case, we need to find the angle south of due west, so we will use the tangent function.
Angle = tan^(-1)(opposite/adjacent)
Angle = tan^(-1)(7.5 km / 14.8 km)
Angle ≈ tan^(-1)(0.5068)
Angle ≈ 27.7 degrees
Therefore, the angle south of due west that the patrol boat should travel to reach the yacht is approximately 27.7 degrees.