A boat sails for 30km on a bearing of 145 degree then 10 km on a bearing of 250 degree, calculate
(a) the distance covered from the starting point
(b) the bearing of the harbour from the ship
145 = 35 degrees south of west
250 = 70 degrees north of west
distance north = -30 sin 35 + 10 sin 70
distance west = 30 cos 35 + 10 cos 70
distance north = -17.2 + 9.4 = -7.8
distance west = 24.6 + 3.4 = 28
tan angle south of west = 7.8/28
angle south of west = 15.6
so compass bearing harbor to ship = 270 -15.6 = 254.4 deg
so PART B is 254.4 - 180 = 74.4 (about east by north)
range from start = sqrt (28^2 + 7.8^2)
(a) Well, the boat sailed 30km in one direction, then 10km in another direction. So, the total distance covered from the starting point is...wait for it...mathematically speaking...drum roll, please...40 kilometers! Ta-da!
(b) Now, the bearing of the harbor from the ship. Hmm, let me think... If the boat was sailing on a bearing of 145 degrees and then changed to a bearing of 250 degrees, then it must have gone off-course a bit, right? So, the harbor is most likely somewhere between 145 and 250 degrees. Let's take a wild guess and say... 200 degrees. Why not? But please note, my guesses are statistically proven to be 48.7% accurate. So, take that with a grain of salt, or a boatload of humor!
To calculate the distance covered from the starting point, we can use the concept of vector addition.
First, let's break down the two legs of the boat's journey into their x and y components.
For the first leg of 30 km on a bearing of 145 degrees:
x component = distance * cos(bearing)
= 30 km * cos(145°)
= -21.73 km
y component = distance * sin(bearing)
= 30 km * sin(145°)
= 20.09 km
For the second leg of 10 km on a bearing of 250 degrees:
x component = distance * cos(bearing)
= 10 km * cos(250°)
= -8.09 km
y component = distance * sin(bearing)
= 10 km * sin(250°)
= -7.68 km
Next, we'll add up the x and y components to find the resulting components:
Total x component = -21.73 km + (-8.09 km) = -29.82 km
Total y component = 20.09 km + (-7.68 km) = 12.41 km
To calculate the distance covered from the starting point, we can use the Pythagorean theorem:
Distance from starting point = sqrt((total x component)^2 + (total y component)^2)
= sqrt((-29.82 km)^2 + (12.41 km)^2)
= sqrt(888.5924 + 153.6481)
= sqrt(1042.2405)
= 32.28 km
Therefore, the distance covered from the starting point is approximately 32.28 km.
To calculate the bearing of the harbor from the ship, we can use the inverse tangent (atan2) function:
Bearing of harbor = atan2(total x component, total y component) + 180°
= atan2(-29.82 km, 12.41 km) + 180°
≈ 111.92° + 180°
= 291.92°
Therefore, the bearing of the harbor from the ship is approximately 291.92°.
To solve this problem, we can use trigonometry and vector addition.
(a) To find the distance covered from the starting point, we need to find the resultant vector by adding the two individual vectors.
1. Convert the bearings to Cartesian coordinates:
For the first part,
Distance = 30 km
Bearing = 145 degrees
To convert this to Cartesian coordinates:
x1 = 30 * cos(145)
y1 = 30 * sin(145)
For the second part,
Distance = 10 km
Bearing = 250 degrees
To convert this to Cartesian coordinates:
x2 = 10 * cos(250)
y2 = 10 * sin(250)
2. Add the Cartesian coordinates together to find the resultant vector:
x_total = x1 + x2
y_total = y1 + y2
3. Calculate the magnitude of the resultant vector using the Pythagorean theorem:
resultant_distance = sqrt(x_total^2 + y_total^2)
(b) To find the bearing of the harbor from the ship, we need to find the angle between the resultant vector and the x-axis.
4. Calculate the angle using the arctan function:
resultant_bearing = arctan(y_total / x_total) in radians
To convert the angle from radians to degrees, multiply by 180/π:
resultant_bearing_degrees = resultant_bearing * (180 / π)
Finally, calculate the distance and bearing.
Note: Make sure to use a calculator that supports radian mode for trigonometric calculations.
Let's calculate the answers:
(a) Distance covered from the starting point:
- Calculate x1 and y1:
x1 = 30 * cos(145) = -18.10
y1 = 30 * sin(145) = 26.77
- Calculate x2 and y2:
x2 = 10 * cos(250) = 2.26
y2 = 10 * sin(250) = -9.40
- Calculate x_total and y_total:
x_total = x1 + x2 = -18.10 + 2.26 = -15.84
y_total = y1 + y2 = 26.77 - 9.40 = 17.37
- Calculate the magnitude of the resultant vector:
resultant_distance = sqrt((-15.84)^2 + (17.37)^2) = 23.20 km
Therefore, the distance covered from the starting point is 23.20 km.
(b) Bearing of the harbor from the ship:
- Calculate the angle in radians:
resultant_bearing = arctan(17.37 / -15.84) = -0.86 radians
- Calculate the angle in degrees:
resultant_bearing_degrees = -0.86 * (180 / π) = -49.37 degrees
Therefore, the bearing of the harbor from the ship is approximately 49.37 degrees (clockwise from the positive x-axis).