two lighthouses are situated such that B is 7 km directly east of lighthouse A. A ship at point P observes that A is due north and that the bearing of B is 46°10'. How far is the ship from A and B ?
According to my sketch:
tan46°10' = 7/AP
AP = 7/ tan46°10' = ....
sin46°10' = 7/BP
BP = 7/sin46°10' = ....
how do you sketch this kind of problem?, that's the reason why i'm having trouble with this ..
first thing, of course, is to draw line AB. B is east of A, so it's horizontal.
A is due north of P, so draw a line south from A and put P somewhere on it.
You can estimate angles, at least to some degree :-) so make PB at about 45 degrees.
To solve this problem, we can use trigonometry. Let's consider point P as the location of the ship.
Step 1: Determining the distance between the ship (P) and lighthouse A
Since the ship observes A due north, we can conclude that the angle between the line connecting P and A (PA) and the north direction is 90°.
Using trigonometry, we can use the sine function to find the length of PA:
sin(90°) = PA / hypotenuse
As the line connecting P and A (PA) is the hypotenuse, we can write:
1 = PA / hypotenuse
Since we don't know the exact length of PA, we can assume it as x:
1 = x / hypotenuse
Simplifying the equation, we get:
hypotenuse = x
So, the distance between the ship (P) and lighthouse A is simply x.
Step 2: Determining the distance between the ship (P) and lighthouse B
To find the distance between the ship (P) and lighthouse B, we need to consider the bearing of B. The bearing of B is given as 46°10'.
The bearing of an object refers to the angle between the line connecting the observer (P) and the object (B) and the north direction, measured in a clockwise direction.
In this case, the angle between the line PB and the north direction is 46°10'.
Step 3: Calculating the distances
We need to use trigonometry again to find the distances.
Based on the information provided, we have a right-angled triangle formed by the points P, B, and A.
Let's consider the angle between PB and the northern direction as theta (θ). Since the bearing of B is 46°10', we can write:
θ = 90° - 46°10' = 43°50'
Using trigonometry, we can use the cosine function to relate the angle θ to the distances PB and PA:
cos(θ) = PB / hypotenuse
Substituting the values, we have:
cos(43°50') = PB / x
To find the distance PB, we need to know the value of x (length of PA). However, we can relate the distances to each other using the given information that B is 7 km directly east of A.
Since B is east of A, the line connecting P and B (PB) is the hypotenuse and line connecting P and A (PA) is the adjacent side of the angle θ in the right-angled triangle.
Using cosine again, we have:
cos(θ) = PB / PA
Substituting the known values, we have:
cos(43°50') = PB / 7 km
Now we have the relationship between PA and PB (PA = x). We can solve this equation to find the value of PB:
PB = 7 km * cos(43°50')
The distance PB is the distance between the ship (P) and lighthouse B.
To find the distance PA (x), we need to solve the equation we obtained in Step 1:
1 = x / hypotenuse
Since hypotenuse = x, we have:
1 = x / x
Therefore, the distance PA (x) is equal to 1 km.
So, the ship is 1 km away from lighthouse A and PB = 7 km * cos(43°50') gives the distance between the ship and lighthouse B.