Two ships sail from port P. One sails 14km due south while the other sails 17km on a bearing of 120 degrees. Calculate to one decimal place the distance between the ships.
I will assume that by a bearing of 120 degrees you mean
S 60° W , so we have a simple application of the cosine law:
From your sketch
x^2 = 14^2 + 17^2 - 2(14)(17) cos 60°
etc
Look at this like a triangle. Port P is the top point of the triangle. One sails 14km straight down (south) the other sails at an angle for 17km(this would be the hypotonus)
Remembering A squared + B squared = C squared
(14*14) + (B*B) =17 *17
196 + (B*B) = 289
B*B = 289-196
B*B = 93
B= square root of 93
B =9.643...….
To find the distance between the ships, we can use the cosine rule. Let's denote the angle between the two ship paths as θ.
Given:
Ship 1 sails 14 km due south.
Ship 2 sails 17 km on a bearing of 120 degrees from the north line.
Step 1: Find the angle θ:
Since Ship 2 is sailing on a bearing of 120 degrees, the angle between the two ship paths can be found by subtracting 120 degrees from 180 degrees.
θ = 180 degrees - 120 degrees
θ = 60 degrees
Step 2: Apply the cosine rule:
The cosine rule states that c^2 = a^2 + b^2 - 2ab * cos(θ), where c represents the distance between the ships, a and b represent the distances traveled by each ship, and θ is the angle between their paths.
Plugging in the given values:
c^2 = 14^2 + 17^2 - 2 * 14 * 17 * cos(60 degrees)
Calculating:
c^2 = 196 + 289 - 476 * cos(60 degrees)
c^2 = 196 + 289 - 476 * 0.5
c^2 = 196 + 289 - 238
c^2 = 247
Step 3: Solve for c:
Taking the square root of both sides, we have:
c = √247
c ≈ 15.7 km
Therefore, the distance between the two ships is approximately 15.7 kilometers.
To solve this problem, we can use the concept of vectors. Let's start by representing the two ships' movements as vectors.
Ship A sails 14 km due south, so its vector can be represented as: 14 km in the direction of 270 degrees (since south is 270 degrees on a compass).
Ship B sails 17 km on a bearing of 120 degrees. To find the vector representation, we need to split this movement into horizontal and vertical components. The horizontal component is given by 17 km multiplied by the cosine of 120 degrees, and the vertical component is given by 17 km multiplied by the sine of 120 degrees.
Horizontal component of ship B = 17 km * cos(120 degrees) = -8.5 km
Vertical component of ship B = 17 km * sin(120 degrees) = 14.7 km
Now, we can find the resultant vector (the vector connecting the two ships) by adding the individual vectors together.
Resultant vector = Vector A + Vector B
To add vectors, we need to add their horizontal and vertical components separately.
Horizontal component of resultant vector = 0 km (since ship A only moves vertically)
Vertical component of resultant vector = 14 km (from ship A) + 14.7 km (from ship B) = 28.7 km
The distance between the ships is given by the magnitude of the resultant vector.
Distance = sqrt((horizontal component)^2 + (vertical component)^2)
Distance = sqrt((0 km)^2 + (28.7 km)^2)
Distance = sqrt(0 + 823.69)
Distance = 28.7 km (rounded to one decimal place)
Therefore, the distance between the ships is approximately 28.7 km.