19-21. A boat sail from A on a bearing of 140 for 40 km to B. Then sails on a bearing of 250 for 80 km to C.
Find distance AC.
Find the bearing of A from C.
Find the bearing of C from A.
make your sketch to see a triangle ABC with angle B = 70°
AC^2 = 40^2 + 80^2 - 2(40)(80)cos70°
= 1600 + 6400 - 2188.9289..
AC = √ .....
= appr 100.94
Use your diagram and this result to find the angles needed
draw a diagram, and then use the law of cosines to find AC
In ∆ABC, B = 70°, so
b^2 = 40^2 + 80^2 - 2(40)(80) cos70°
if A is at (0,0) then
B is at (25.71,-30.64)
C is at (-49.46,-58.00)
so the bearing of A from C is θ, where
tanθ = 58/49.46
making the bearing of C from A θ+180°
To find the distance AC, we can use the law of cosines.
1. Let's calculate the distance AB using the cosine rule:
AB^2 = 40^2 + 40^2 - 2 * 40 * 40 * cos(180 - 140)
AB^2 = 1600 + 1600 - 3200 * cos(40)
AB^2 = 3200 - 3200 * cos(40)
AB = √(3200 - 3200 * cos(40))
2. Now, let's calculate the distance BC using the cosine rule:
BC^2 = 80^2 + 80^2 - 2 * 80 * 80 * cos(180 - 250)
BC^2 = 6400 + 6400 - 12800 * cos(70)
BC^2 = 12800 - 12800 * cos(70)
BC = √(12800 - 12800 * cos(70))
3. Finally, to find the distance AC, we can use the Pythagorean theorem:
AC^2 = AB^2 + BC^2
AC = √(AB^2 + BC^2)
Now, let's calculate the bearing of A from C:
4. To find the bearing of A from C, we can use the tangent formula:
tan(bearing of A from C) = (BC * sin(250 - 180)) / (AB + BC * cos(250 - 180))
bearing of A from C = arctan((BC * sin(70)) / (AB - BC * cos(70)))
And finally, let's calculate the bearing of C from A:
5. To find the bearing of C from A, we can use the tangent formula:
tan(bearing of C from A) = (AB * sin(140)) / (BC - AB * cos(140))
bearing of C from A = arctan((AB * sin(140)) / (BC - AB * cos(140)))
Please let me know if you need any further assistance with these calculations.
To find the distance AC, we can use the formula for finding the distance between two points using their coordinates. Let's assume that point A is the origin (0, 0) on a coordinate plane.
Since the boat sails from A on a bearing of 140 degrees for 40 km to B, we can determine the coordinates of point B using trigonometry. The horizontal distance traveled from A to B is given by the formula: horizontal distance = distance * cos(bearing). In this case, the horizontal distance from A to B is 40 km * cos(140°).
To find the vertical distance traveled from A to B, we use the formula: vertical distance = distance * sin(bearing). In this case, the vertical distance from A to B is 40 km * sin(140°).
Now, we can find the coordinates of point B as follows:
x-coordinate of B = x-coordinate of A + horizontal distance = 0 + 40 km * cos(140°)
y-coordinate of B = y-coordinate of A + vertical distance = 0 + 40 km * sin(140°)
Now, the boat sails from B on a bearing of 250 degrees for 80 km to reach point C. Similarly, we can calculate the coordinates of point C using the formulas mentioned earlier:
x-coordinate of C = x-coordinate of B + horizontal distance = x-coordinate of B + 80 km * cos(250°)
y-coordinate of C = y-coordinate of B + vertical distance = y-coordinate of B + 80 km * sin(250°)
Once we have the coordinates of point C, we can find the distance AC using the distance formula between two points:
distance AC = square root of [(x-coordinate of C - x-coordinate of A)^2 + (y-coordinate of C - y-coordinate of A)^2]
For the bearing of A from C, we can use the formula: bearing = arctan((y-coordinate of C - y-coordinate of A) / (x-coordinate of C - x-coordinate of A))
Similarly, for the bearing of C from A, we can use the formula: bearing = arctan((y-coordinate of A - y-coordinate of C) / (x-coordinate of A - x-coordinate of C))
By substituting the calculated values into the respective formulas, we can find the distance AC, the bearing of A from C, and the bearing of C from A.