Abbey steers her boat in a North Easterly direction for 15 s, East for 6 s and then stops to avoid hitting a duck. If her boat travels at a constant speed of 10 m/min during this time, what is the distance and compass bearing of Abbey’s boat from the starting point when it stops? Distances are to be rounded to 2 decimal places and angle sizes to 1 decimal place.
t = 15s @ 45Deg. + 6s. @ 0 Deg.
X = 15*cos45 + 6 = 16.61 s.
Y = 15*sin45 = 10.61 s.
tanA = Y/X = 10.61 / 16.61 = 0.63877.
A = 32.57 Deg.
t =X / cos32.57 = 16.61 / cos32.57 = 19.71 s @ 32.57 Deg..
V = 10m/min = 10m/60s = 0.1667m/s.
d = V*t = 0.1667m/s * 19.71s = 3.29m
@32.57 Deg.
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Thanks for the answer but can this be worked out using the cosine and sine rules
Like the Cosine rule:
b^2=a^2+c^2-2ac cosB
or the Sine rule
a/sinA = b/sinB = c/sinC
To find the distance and compass bearing of Abbey's boat from the starting point, we can break down her movement into two components: the North-South component and the East-West component.
1. Calculate the North-South component:
Abbey steers in a North-Easterly direction for 15 seconds, which means she is traveling at a 45-degree angle between North and East. To find the North-South component, we need to calculate the distance traveled in the North direction.
Distance traveled in the North direction = (Speed) x (Time)
= 10 m/min x (15 s / 60 s/min) (converting seconds to minutes)
= 2.5 m
2. Calculate the East-West component:
Abbey steers to the East for 6 seconds. This means she is traveling in the East direction.
Distance traveled in the East direction = (Speed) x (Time)
= 10 m/min x (6 s / 60 s/min) (converting seconds to minutes)
= 1 m
3. Calculate the overall distance:
To find the overall distance from the starting point, we need to calculate the straight-line distance using the Pythagorean theorem.
Overall distance = √(North-South component)^2 + (East-West component)^2
= √(2.5 m)^2 + (1 m)^2
= √6.25 + 1
= √7.25
= 2.69 m (rounded to 2 decimal places)
4. Calculate the compass bearing:
To find the compass bearing, we need to calculate the angle between the boat's position and the North direction. We can use trigonometry to find this angle.
Angle = arctan((North-South component) / (East-West component))
Angle = arctan(2.5 m / 1 m)
= arctan(2.5)
To convert this angle to compass bearing, we need to add or subtract an appropriate value based on the quadrant in which the angle lies. In this case, the angle is positive and less than 90 degrees, indicating that the boat is east of the starting point.
Compass bearing = 90 degrees - Angle
= 90 degrees - arctan(2.5)
= 90 degrees - 67.4 degrees
= 22.6 degrees (rounded to 1 decimal place)
Therefore, the distance of Abbey's boat from the starting point is 2.69 m, and the compass bearing is 22.6 degrees.