Two boats A and B leave port P at the same time. Boat A sails 30km in the North east direction and boat B sails 20km on a bearing of 300 degrees. What is the distance between the two boats and what is the bearing of boat A from boat B
Why are you using two different directional notations?
I will convert them to standard trig notation.
NE ----> 45°
"bearing" of 300 ---> 150°
angle between the two arms = 105° (make a sketch to see)
now if the distance between them is d , then
d^2 = 20^2 + 30^2 - 2(20)(30)cos105
= 1300 - 1200(-.25882)
d = approx 40.13
find one of the other angles using the Sine Law, and you can then
find all the angles in your sketch
To find the distance between the two boats, we can use the Pythagorean theorem. Since Boat A sails 30km in the northeast direction, we can split it into two legs: 30km eastward and 30km northward.
The distance travelled by Boat A is the hypotenuse of a right-angled triangle, with the two legs being 30km each. Using the Pythagorean theorem, we have:
Distance^2 = (30km)^2 + (30km)^2
Distance^2 = 900km^2 + 900km^2
Distance^2 = 1800km^2
Distance = √1800km
Distance ≈ 42.43km
Now, let's find the bearing of Boat A from Boat B. The bearing is measured clockwise from the north direction, so we need to calculate the angle between the direction of Boat B and the direction of Boat A.
To do this, we start by finding the bearing of Boat B. Boat B sails on a bearing of 300 degrees, which is measured clockwise from the north direction. Since the bearing is given with respect to north, we need to find the angle between the east direction and the bearing.
Since east is 90 degrees clockwise from north, and the bearing is 300 degrees clockwise from north, the angle between east and the bearing is:
Angle = 300 degrees - 90 degrees
Angle = 210 degrees
To find the bearing of Boat A from Boat B, we subtract the angle of Boat B from the bearing of Boat A. The bearing of Boat A is 45 degrees, measured clockwise from north-east. So:
Bearing = 45 degrees - 210 degrees
Bearing = -165 degrees
However, bearings are typically measured between 0 and 360 degrees. To convert the bearing to a positive value, we add 360 degrees:
Bearing = -165 degrees + 360 degrees
Bearing = 195 degrees
Therefore, the distance between the two boats is approximately 42.43km, and the bearing of Boat A from Boat B is 195 degrees.
To find the distance between the two boats, we can consider them forming a right triangle. Boat A's 30km north-east (NE) direction can be divided into two components: one northward and one eastward.
Using the Pythagorean theorem, we can calculate these components. In a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). In this case, a represents the northward distance and b represents the eastward distance.
Given that Boat A sails 30km in the NE direction, we can determine the northward component (a) and the eastward component (b). Since it is a right-angled triangle, both components will have the same magnitude.
Using basic trigonometry, we can calculate the magnitude of these components.
The northward component (a) can be calculated as: a = 30km * sin(45°) (since NE is 45 degrees from north)
Similarly, the eastward component (b) can be calculated as: b = 30km * cos(45°) (since NE is 45 degrees from east)
a = 30km * 0.707 ≈ 21.21km
b = 30km * 0.707 ≈ 21.21km
Now that we know the northward (a = 21.21km) and eastward (b = 21.21km) components for Boat A, we can use them to calculate the distance between the two boats.
The distance between the two boats is the hypotenuse (c) of the right triangle formed. We can use the Pythagorean theorem to find it:
c = sqrt(a^2 + b^2)
c = sqrt((21.21km)^2 + (21.21km)^2)
c = sqrt(449.57km^2 + 449.57km^2)
c = sqrt(2 * 449.57km^2)
c = sqrt(898.114km^2)
c ≈ 29.96km
Hence, the distance between the two boats is approximately 29.96km.
To find the bearing of boat A from boat B, we can use the concept of bearings, which is based on compass directions.
The bearing of an object is defined as the angle between the north direction and the line from the observer to the object. It is measured clockwise from the north.
In this case, we can find the bearing of boat A from boat B by calculating the angle formed between their eastward components.
The eastward component of Boat A is 21.21km, and the eastward component of Boat B is 20km (given in the question).
Since the bearing of an object is measured clockwise from the north, we need to find the angle between the eastward component of Boat A (21.21km) and the eastward component of Boat B (20km).
To find this angle, we can use trigonometric functions.
Using trigonometry, the angle θ can be calculated as:
θ = arctan((21.21km - 20km)/29.96km)
θ = arctan(1.21km/29.96km)
θ ≈ 2.43°
Hence, the bearing of boat A from boat B is approximately 2.43 degrees clockwise from the north.