Three oil platforms, Alpha, Beta and Gamma
are situated in the North Sea.Beta is 12 km from Alpha and is on a bearing of 145 degrees from Alpha.Gamma is 16 km from Alpha and is on a bearing of 220 degrees from Alpha.Find the distance of Beta from Gamma.
(Not sure how to approach this question)
draw a diagram. Use the law of cosines, since you have two sides and the angle between them.
To find the distance of Beta from Gamma, we can use the cosine rule. The cosine rule states that in a triangle with sides a, b, and c, and angles A, B, and C opposite to the respective sides:
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, let's call the distance from Alpha to Beta as side a, the distance from Alpha to Gamma as side b, and the distance from Beta to Gamma as side c.
We already know that a = 12 km and b = 16 km. We need to find c.
To find angle C, we can use the bearing information. The bearing from Alpha to Beta is 145 degrees, while the bearing from Alpha to Gamma is 220 degrees. The angle between these two bearings is the angle at Beta (opposite to side c).
To find angle C, we subtract the bearing of Alpha to Gamma (220 degrees) from the bearing of Alpha to Beta (145 degrees):
C = 220 - 145 = 75 degrees
Now we can apply the cosine rule:
c^2 = a^2 + b^2 - 2ab*cos(C)
c^2 = 12^2 + 16^2 - 2 * 12 * 16 * cos(75)
c^2 = 144 + 256 - 384 * 0.2588
c^2 = 400 - 99.3312
c^2 = 300.6688
c ≈ 17.32 km
Therefore, the distance of Beta from Gamma is approximately 17.32 km.
To find the distance between Beta and Gamma, we can use the concept of vectors.
First, let's represent the positions of Alpha, Beta, and Gamma as vectors. Let the vectors A, B, and G represent the positions of Alpha, Beta, and Gamma, respectively.
Given that Beta is 12 km from Alpha and on a bearing of 145 degrees from Alpha, we can determine the x and y coordinates of B relative to A.
Using trigonometry, we can find that the x-coordinate of B relative to A is 12 * cos(145 degrees) and the y-coordinate is 12 * sin(145 degrees). Therefore, vector B can be represented as B = (12 * cos(145 degrees), 12 * sin(145 degrees)).
Similarly, given that Gamma is 16 km from Alpha and on a bearing of 220 degrees from Alpha, we can determine the x and y coordinates of G relative to A.
Using trigonometry, we can find that the x-coordinate of G relative to A is 16 * cos(220 degrees) and the y-coordinate is 16 * sin(220 degrees). Therefore, vector G can be represented as G = (16 * cos(220 degrees), 16 * sin(220 degrees)).
Now, to find the distance between Beta and Gamma, we can use the formula for the Euclidean distance between two points in a Cartesian coordinate system:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2).
In this case, x1 and y1 are the coordinates of Beta (B) relative to Alpha (A), and x2 and y2 are the coordinates of Gamma (G) relative to Alpha (A).
So, the distance between Beta and Gamma is:
distance = sqrt((16 * cos(220 degrees) - 12 * cos(145 degrees))^2 + (16 * sin(220 degrees) - 12 * sin(145 degrees))^2).
Evaluating this expression will give us the distance between Beta and Gamma.