A boat sails 4km on a bearing of 038 and then 5km on a bearing of 067. a. how far is the boat from its starting point? b. Calculate the bearing of the boat from its starting point.

Question

A boat sails 4km on a bearing of 038 and then 5km on a bearing of 067. a. how far is the boat from its starting point?  b. Calculate the bearing of the boat from its starting point.

Sarah · Accepted Answer

SOME SOLVED THIS IN 2016

HERE IS THEIR WORK:
simplest way is using vectors.
resultant vector
= (4cos38,4sin38) + (5cos67,5sin67)

= (3.152.., 2.4626..) + (1.9536..., 4.6025..)
= (5.1056.., 7.065...)

distance = √(5.1056..^2 + 7.065.^2)
= appr 8.717 correct to 3 decimals

bearing:
tanØ = 7.065../5.1056..
Ø = 54.146°

OR

Using basic geometry:
make a sketch, on mine I have a triangle with sides 4 and 5 with an angle of 151° between them
by the cosine law:
d^2 = 4^2 + 5^2 - 2(4)(5)cos151
= 8.717 , just like above

let the angle opposite the 5 km side be Ø
by the sine law:
sinØ/5 = sin 151/8.717
sinØ = .27808..
Ø = 16.146
bearing = Ø + 38°
= 16.146+38
= 54.146° , again, just as above

Sarah · Answer

I couldn't post the link - it doesn't allow me. But, you can copy and paste your question in google or on here and it will direct you to the Jiksha page for this question which was answered in 2016, as well as in 2021 by 2 different people!

Faith · Answer

Calculation and formula

Harriet · Answer

by the cosine law:
d^2 = 4^2 + 5^2 - 2(4)(5)cos151
= 8.717 , just like above

let the angle opposite the 5 km side be Ø
by the sine law:
sinØ/5 = sin 151/8.717
sinØ = .27808..
Ø = 16.146
bearing = Ø + 38°
= 16.146+38
= 54.146° , again, just as above

Bot · Answer

The calculation above uses the cosine law and sine law to find the distance and bearing of a boat that sails 4km on a bearing of 038 and then 5km on a bearing of 067.

The cosine law states that in a triangle with sides a, b, and c and opposite angles A, B, and C, the following equation holds true:

c^2 = a^2 + b^2 - 2abcosC

Using this formula, the distance d between the starting point and final point of the boat is calculated. In this case, a = 4km, b = 5km, and C = 151 degrees (180 - 67).

d^2 = 4^2 + 5^2 - 2(4)(5)cos151
= 8.717 km

To find the bearing of the boat from its starting point, the sine law is used. The sine law states that in a triangle with sides a, b, and c and opposite angles A, B, and C, the following equation holds true:

a/sinA = b/sinB = c/sinC

Using this formula, the angle Ø opposite the 5km side is found.

sinØ/5 = sin 151/8.717
sinØ = .27808..
Ø = 16.146 degrees

Finally, the bearing is calculated by adding 38 degrees (the initial bearing) to Ø.

bearing = Ø + 38°
= 16.146+38
= 54.146°