"A ship leaves port on a bearing of 34.0 degrees and travels 10.4 mi. The ship then turns due east and travels 4.6 mi. How far is the ship from port, and what is its bearing from port?"
I would like to know how the actual steps in finding the answer please!
just start drawing your lines, and figure out how far each leg of the trip takes the ship. Start at (0,0)
go 10.4 mi at N34°E puts you at
(10.4*sin34°,10.4*cos34°) = (5.82,8.62)
Now go E for 4.6, and that adds (4.6,0) to the displacement. Now you are at
(10.42,8.62)
The new bearing θ is given by
tanθ = 10.42/8.62 = 1.2088
θ = 50.4°
distance is sqrt(182.88) = 13.52
or, 13.52mi at N50.4°E
not an answer, but steve, how did you get sqrt of 182.88? just wondering.
square 10.42 and 8.62 and add to get 182.88
To find the distance and bearing of the ship from the port, we can use trigonometry and vector addition.
Step 1: Draw a diagram
Start by drawing a diagram representing the situation. Mark the port as the starting point (point A) and the final position of the ship as point B.
Step 2: Calculate the x and y components of the ship's displacement
Since the ship first travels on a bearing of 34.0 degrees, its displacement can be divided into horizontal (x) and vertical (y) components. We can use the following trigonometric formulas to find the components:
x = 10.4 mi * cos(34.0 degrees)
y = 10.4 mi * sin(34.0 degrees)
Step 3: Calculate the final position of the ship
After the ship turns due east and travels 4.6 miles, its displacement can be represented as a vector added to the previous displacement. Let's call this vector BC.
The x component of BC is 4.6 mi, and the y component remains the same as it is traveling due east. So the final position of the ship (point C) can be calculated as follows:
C(x) = B(x) + BC(x)
C(y) = B(y) + BC(y)
Step 4: Find the distance and bearing of the ship from the port
The distance between the port (A) and the ship's final position (C) can be calculated using the distance formula:
Distance = √((C(x) - A(x))^2 + (C(y) - A(y))^2)
The bearing of the ship from the port can be found using the inverse tangent (arctan) function:
Bearing = arctan((C(y) - A(y)) / (C(x) - A(x)))
Plug in the calculated values to get the final answers.