Starting at point A, a ship sails 57 km on a bearing of 188°, then turns and sails 37 km on a
bearing of 330°. Find the distance of the ship from point A.
the best way I could think of is break the distances into horizontal (x) and vertical (x) components and use Pythagorean theorem to find the distance of the ship from point A.
x= 57cos188 + 37cos330
y= 57sin188 + 37sin330
x^2 + y^2 = c^2
find c.
C= -1293,9
To solve this problem, we can break it down into two steps:
Step 1: Find the coordinates of the ship after the first leg of the journey
Step 2: Calculate the distance between point A and the final coordinates of the ship
Step 1: Finding the coordinates after the first leg
To do this, we'll use basic Trigonometry and the formula for finding the coordinates of a point using distance and bearing.
Let's define point A as (x, y), where x and y are the coordinates.
On the first leg, the ship sails 57 km on a bearing of 188°. This means it travels in a southwesterly direction.
Using Trigonometry, we can calculate the change in x and y coordinates.
Change in x = 57 km * sin(188°)
Change in y = 57 km * cos(188°)
To find the new coordinates, we need to subtract the change in x and y from the original coordinates.
New x = x - (57 km * sin(188°))
New y = y - (57 km * cos(188°))
Step 2: Calculating the distance between point A and the final coordinates
After the ship completes the second leg on a bearing of 330°, we can use the Pythagorean theorem to calculate the distance between the final coordinates and point A.
The change in x = 37 km * sin(330°)
The change in y = 37 km * cos(330°)
Final x = New x - (37 km * sin(330°))
Final y = New y - (37 km * cos(330°))
To find the distance, we can use the Pythagorean theorem:
Distance = √((Final x - x)^2 + (Final y - y)^2)
By substituting the values, we can calculate the distance.
To find the distance of a ship from point A after sailing on different bearings, we can break down the journey into two separate legs.
First, let's calculate the coordinates of the ship at the end of each leg using trigonometry.
Starting at point A, we sail 57 km on a bearing of 188°. This means the ship moves 57 km in the direction 188° clockwise from North.
To calculate the change in latitude (Δ𝑙𝑎𝑡) and longitude (Δ𝑙𝑜𝑛) at point A, we use trigonometric functions:
Δ𝑙𝑎𝑡 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 × 𝑠𝑖𝑛(𝑏𝑒𝑎𝑟𝑖𝑛𝑔)
Δ𝑙𝑜𝑛 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 × 𝑐𝑜𝑠(𝑏𝑒𝑎𝑟𝑖𝑛𝑔)
Solving for Δ𝑙𝑎𝑡 and Δ𝑙𝑜𝑛:
Δ𝑙𝑎𝑡 = 57 × 𝑠𝑖𝑛(188°)
Δ𝑙𝑎𝑡 ≈ 57 × -0.574
Δ𝑙𝑎𝑡 ≈ -32.758 km
Δ𝑙𝑜𝑛 = 57 × 𝑐𝑜𝑠(188°)
Δ𝑙𝑜𝑛 ≈ 57 × -0.818
Δ𝑙𝑜𝑛 ≈ -46.626 km
Now, we can calculate the coordinates of the ship after the first leg:
New latitude = latitude of A + Δ𝑙𝑎𝑡
New longitude = longitude of A + Δ𝑙𝑜𝑛
Next, the ship turns and sails 37 km on a bearing of 330° from its new position after the first leg.
Similarly, we calculate the change in latitude (Δ𝑙𝑎𝑡) and longitude (Δ𝑙𝑜𝑛) caused by the second leg:
Δ𝑙𝑎𝑡 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 × 𝑠𝑖𝑛(𝑏𝑒𝑎𝑟𝑖𝑛𝑔)
Δ𝑙𝑜𝑛 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 × 𝑐𝑜𝑠(𝑏𝑒𝑎𝑟𝑖𝑛𝑔)
Solving for Δ𝑙𝑎𝑡 and Δ𝑙𝑜𝑛:
Δ𝑙𝑎𝑡 = 37 × 𝑠𝑖𝑛(330°)
Δ𝑙𝑎𝑡 ≈ 37 × -0.571
Δ𝑙𝑎𝑡 ≈ -21.127 km
Δ𝑙𝑜𝑛 = 37 × 𝑐𝑜𝑠(330°)
Δ𝑙𝑜𝑛 ≈ 37 × -0.821
Δ𝑙𝑜𝑛 ≈ -30.377 km
Now, we can calculate the final coordinates of the ship:
New latitude = latitude after the first leg + Δ𝑙𝑎𝑡
New longitude = longitude after the first leg + Δ𝑙𝑜𝑛
Finally, we can find the distance of the ship from point A using the distance formula:
Distance = √((latitude after the first leg - latitude of A)² + (longitude after the first leg - longitude of A)²)
Substituting the calculated values, we can find the distance of the ship from point A.