A ship is 60 miles west and 91 miles south of the harbor.
a) What bearing should the ship take to sail directly to the harbor? (Round answer to nearest tenth of degree.)
b) What is the direct distance to the harbor?
TanA = 60/91 = 0.65934, A = 33.4o. E. of N.
a) Well, since the ship is feeling a bit lost, it can always rely on its trusty compass. To sail directly to the harbor, the ship should take a bearing of about 243.4 degrees. Don't worry ship, the harbor will soon be in sight!
b) As for the direct distance to the harbor, we can put on our mathematical sailor hats and calculate it using the Pythagorean theorem. The ship is 60 miles west and 91 miles south, so if we draw a right triangle, the distance to the harbor is like the hypotenuse. Using the magical powers of math, we can find that the direct distance to the harbor is approximately 109.8 miles. Ahoy, harbor! Here we come!
a) To find the bearing the ship should take to sail directly to the harbor, we can use trigonometry.
First, we need to find the angle between the line connecting the ship and the harbor, and the north-south line.
Let's label the ship as point A (60 miles west and 91 miles south of the harbor), and the harbor as point B.
The distance traveled north-south is 91 miles, and the distance traveled east-west is 60 miles.
The angle can be found using the tangent function:
tan(angle) = (Distance traveled north-south) / (Distance traveled east-west)
tan(angle) = 91 / 60
Using a calculator, we can find the inverse tangent (arctan) of this value:
angle ≈ arctan(91/60) ≈ 55.59 degrees
However, this gives us the bearing from the harbor to the ship. To find the bearing the ship should take to sail directly to the harbor, we need to add 180 degrees to this bearing, since the direction will be opposite.
Therefore, the bearing the ship should take to sail directly to the harbor is approximately:
180 + 55.59 ≈ 235.6 degrees (rounded to the nearest tenth of a degree)
b) To find the direct distance to the harbor, we can use the Pythagorean theorem.
The direct distance (hypotenuse) can be found as the square root of the sum of the squares of the distances traveled north-south and east-west:
Direct distance = √(91^2 + 60^2)
Using a calculator, we can find:
Direct distance ≈ √(8281 + 3600) ≈ √11881 ≈ 109.1 miles (rounded to the nearest tenth of a mile)
To find the bearing the ship should take to sail directly to the harbor, we can use trigonometry. The bearing is the angle between the line connecting the ship to the harbor and the north direction. Here's how we can calculate it:
a) Find the angle between the line connecting the ship and the harbor, and the east direction.
To get the angle, we need to use the tangent function. The tangent of an angle is equal to the length of the opposite side (south in this case) divided by the length of the adjacent side (west in this case).
Let's call the angle we are looking for theta. So we have:
tan(theta) = (opposite/adjacent) = (91/60)
Next, we can solve for theta by taking the inverse tangent (arctan) of (91/60):
theta = arctan(91/60)
Using a calculator, the value of theta is approximately 56.3 degrees.
However, this is the angle with respect to the east direction. To find the bearing with respect to the north direction, we need to subtract this angle from 90 degrees (since north is 90 degrees from east):
Bearing = 90 - theta = 90 - 56.3 ≈ 33.7 degrees.
Therefore, the ship should take a bearing of approximately 33.7 degrees to sail directly to the harbor.
b) To find the direct distance to the harbor, we can use the Pythagorean theorem. The direct distance is the hypotenuse of the right triangle formed by the ship's location and the harbor.
Using the given information, we have:
Distance = sqrt((west)^2 + (south)^2)
Distance = sqrt((60^2) + (91^2))
Using a calculator, the distance is approximately 109.4 miles.
So, the direct distance to the harbor is approximately 109.4 miles.