A ship travels 700 nautical miles on a bearing of 102 degrees true...?

Question

and then 123 nautical miles on a bearing of 131 degrees true. How far south of its starting point is it?

Can someone please explain to me how I work this question out?!

Answer 1

Grab any chart of a scale that will allow you to draw a line 700 miles long. Make a mark in an empty space. Go to the nearest compass rose and align your parallel ruler with 102 degrees true (the outer ring).

'Walk' the parallel rulers down to the mark and make a long line using the ruler as a guide. Now get a set of dividers. Use the scale at the bottom of the chart and widen the dividers to 100km. Start at the mark you made and 'walk' the dividers down that line 7 times. Make a mark.

Now repeat it all for the next part for 131 degrees and 123 miles. Make a second mark. Draw a line parallel with the longitude marks to a point directly under your first mark. That's the distance South of your start point. Use the Dividers to measure it against the scale.

Well... that's how I do it on a ship, anyway.

Another way is by using geometry - create a triangle - a line from the start point drawn down at an angle of 102 degrees from vertical and using a scale of 1cm to 100k gives you a line 7cm's long. That's one side of the triangle. From that point draw a horizontal line back and directly under the start point. That's the 2nd side of the triangle. Now join up the bottom to the start point to give you the third side - which is the distance South. Measure it in cm's and multiply by 100 - and there's your answer.

Now we'll wait until a brainiac tell us the easy way!

Steve

Answer 2

The key to the problem  is the definition of degrees true. See the link below.

Answer 3

South = 700sin(102 - 90) + 123sin(131 - 90)

= 700sin(12°) + 123sin(41°) ≈ 226.233344 nautical miles south

___________

You are also quite a ways east of your starting point but you didn't ask about that.

Answer 4

The hardest part is drawing a diagram.

I am using 3 figure bearings ie measuring clockwise from north.

Draw ∆ ABD (not to scale) such that :-

AD is horizontal

DB is vertical

B lies below D

/_ A = 12° = /_BAD = 12°

AB = 700

/_B = 78°

Extend  DB to a point F below B such that /_CBF = 49°

C lies to right of F (CF is horizontal)

Now have ∆ ABC in which :-

AB = 700

BC = 123

/_ ABC = 151°

Using cosine rule :-

AC² = 700² + 123² - (2) (700)(123) cos 151°

AC = 810 miles

Using sine rule in ∆ ABC :-

AC / sin 151° = BC / Sin A

sin A = BC sin 151° / AC

sin A = 123 sin 151° / 810

A = 4.22° = /_CAB

/_CAD = 16.22°

Draw ∆ AEC right angled at E

E lies below A

EC is horizontal

/_EAC = 73.8°

cos 73.8° = AE / 810

AE = cos 73.8° x 810

AE = 226 nautical miles (south of starting point)

Easy to make a mistake in this when trying to work without an on screen diagram.

Fingers crossed !

Answer 5

When the ship travels on a bearing of 102 , its angle to a vertical line (the line which is from true Nto true S is 180-102=78 gegrees.When it travels to a bearing of 131 then its angle is 180-131=49 degrees.

The distance Sought of starting point is obtained as:

700 x Cos 78 + 123 Cos 49=700 x 0.208 + 123 x 0.656=226.288 miles.

If we want to know how far it is east of starting point , then we have:

700 Sin 78+ 123 Sin 49=700 x 0.978+123 x 0.755=777.456 miles

Finally if we want to know how far it is from the starting point, we have:

D^2=(226.288)^2 + (777.456)^2

D^2=655644/09088

D=809.7185 miles