from a point x a boat sails 6 km on a bearing of 038 degrees to a point y.it then sails 7 km from y on a bearing of 067 degrees to a point z.calculate the;
a.distance XZ correct to two decimal places
b.bearing of Z from X correct to the nearest degree
boats do not sail on bearings. They sail on headings.
angle XYZ is 151°
So, using the law of cosines, the distance XZ is
z^2 = 6^2 + 7^2 - 2*6*7*cos151°
If you draw the diagram, you will be able to figure all the angles needed to find the desired bearing.
To calculate the distance XZ, we can use the cosine rule. The cosine rule states that in a triangle with side lengths a, b, and c, and with angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
Let's calculate the distance XZ using this formula:
Step 1: Draw a diagram to visualize the problem.
Based on the given information, point X is the starting point and point Z is the final destination. The boat sails 6 km from X to Y and then 7 km from Y to Z.
Y
/
/
X---Z
Step 2: Calculate the distance XY using the cosine rule.
Let a = 6 km, b = 7 km, and C = 38 degrees.
c^2 = a^2 + b^2 - 2ab * cos(C)
c^2 = 6^2 + 7^2 - 2 * 6 * 7 * cos(38)
c^2 = 36 + 49 - 84 * cos(38)
c^2 = 85.34
Taking the square root of both sides, we get:
c = sqrt(85.34)
c ≈ 9.24
So, the distance XY is approximately 9.24 km.
Step 3: Now, calculate the distance XZ using Pythagoras' theorem.
Since XY and YZ form a right triangle, we can use Pythagoras' theorem to find the distance XZ.
Using the given distances, we have:
XZ^2 = XY^2 + YZ^2
XZ^2 = 9.24^2 + 7^2
XZ^2 = 85.34 + 49
XZ^2 ≈ 134.34
Taking the square root of both sides, we get:
XZ = sqrt(134.34)
XZ ≈ 11.59 km
So, the distance XZ is approximately 11.59 km (correct to two decimal places).
To calculate the bearing of point Z from point X, we can use the tangent rule. The tangent rule states that in a triangle with sides a, b, and angle C opposite side c, the following equation holds:
tan(C) = (sin(A) * cos(c) - cos(A) * sin(c)) / (cos(A) * cos(c) + sin(A) * sin(c))
Let's calculate the bearing of Z from X using this formula:
Step 4: Calculate the bearing of Z from X.
Let A be the bearing of Y from X (038 degrees) and let c be the distance XY (9.24 km).
tan(C) = (sin(A) * cos(c) - cos(A) * sin(c)) / (cos(A) * cos(c) + sin(A) * sin(c))
tan(C) = (sin(38) * cos(9.24) - cos(38) * sin(9.24)) / (cos(38) * cos(9.24) + sin(38) * sin(9.24))
tan(C) = (0.615661 * 0.974429 - 0.78801 * 0.224951) / (0.78801 * 0.974429 + 0.615661 * 0.224951)
tan(C) ≈ -0.135959
Taking the inverse tangent of -0.135959 using a calculator, we get:
C ≈ -7.80 degrees
Since the bearing is measured clockwise from the north, we need to add 180 degrees to the result to obtain the bearing in the usual format:
Bearing of Z from X ≈ -7.80 + 180 ≈ 172 degrees
So, the bearing of point Z from point X is approximately 172 degrees (rounded to the nearest degree).
To find the distance XZ and the bearing of Z from X, we can use trigonometry and the concept of vectors.
a. Distance XZ:
To find the distance XZ, we can break down the path into two components - one from X to Y and the other from Y to Z.
Using the first leg of the journey from X to Y, we can find the horizontal and vertical components of the distance traveled. Let's call the horizontal component dX and the vertical component dY.
dX = 6 km * cos(38°)
dY = 6 km * sin(38°)
Now let's calculate the position of Y relative to X:
X-coordinate of Y = X-coordinate of X + dX = 0 + dX
Y-coordinate of Y = Y-coordinate of X + dY = 0 + dY
Next, we move from Y to Z. Similarly, let's call the horizontal and vertical components of this leg of the journey dY2 and dZ.
dY2 = 7 km * cos(67°)
dZ = 7 km * sin(67°)
Now let's calculate the position of Z relative to Y:
X-coordinate of Z = X-coordinate of Y + dY2 = dX + dY2
Y-coordinate of Z = Y-coordinate of Y + dZ = dY + dZ
Finally, using the coordinates of X and Z, we can find the distance between them (XZ) using the distance formula:
XZ = sqrt((X-coordinate of Z - X-coordinate of X)^2 + (Y-coordinate of Z - Y-coordinate of X)^2)
Substituting the values we calculated, we get:
X-coordinate of Z = dX + dY2 = 6 km * cos(38°) + 7 km * cos(67°)
Y-coordinate of Z = dY + dZ = 6 km * sin(38°) + 7 km * sin(67°)
XZ = sqrt((6 km * cos(38°) + 7 km * cos(67°) - dX)^2 + (6 km * sin(38°) + 7 km * sin(67°) - dY)^2)
Evaluating this expression will give us the distance XZ.
b. Bearing of Z from X:
To find the bearing of Z from X, we can use the arctan function. The bearing (θ) can be calculated using the following formula:
θ = arctan((Y-coordinate of Z - Y-coordinate of X) / (X-coordinate of Z - X-coordinate of X))
Evaluating this expression will give us the bearing of Z from X.