Bearings Problem Solving Question: From a lighthouse two boats can be sighted. Boat 1 on a bearing of 60° at a distance of 2km and Boat 2 on a bearing of 340° at a distance of 3km.
a) If Boat 1 is in need of assistance, what bearing would Boat 2 need to travel on to reach it?
b) Assuming Boat 1 does not move, how far would Boat 2 have to travel?

9514 1404 393

  a) 123.4°

  b) 3.3 km

Step-by-step explanation:

In (N, E) coordinates, Boat 1 is located relative to the lighthouse at ...

  2(cos(60°), sin(60°)) = (1.0, 1.732) . . . . . km

Boat 2 is located at ...

  3(cost(340°), sin(340°)) = (2.819, -1.026) . . . . . km

So, relative to Boat 2, Boat 1 is located at ...

  (1, 1.732) -(2.819, -1.026) = (-1.819, 2.758) . . . . . km

This is at a bearing of ...

  arctan(2.758/-1.819) = 123.4°

and a distance of ...

  √(1.819² +2.758²) ≈ 3.304 . . . . . km

a) Boat 2 would need to travel on a bearing of 123.4°.

b) Boat 2 would need to travel 3.3 km.

_____

Additional comment

We use (North, East) coordinates so we don't have to do the mental gyrations necessary to translate to and from (E, N) coordinates. The latter would correctly orient a map relative to the X-Y Cartesian plane. The former are equivalent to flipping the map over and looking at it from the back side. This is a "rigid transformation", so angles and distances remain unchanged.

step-by-step explanation:

12: 00 p.m. to 7: 00 p.m. is a total of 7 hours.

4: 00 p.m. to 7: 00 p.m. is a total of 3 hours.

this means that from 4: 00 p.m to 7: 00 p.m. this represents 3/7 of what was lost from 12: 00 p.m. to 7: 00 p.m.

after knowing this, after doing 11,600 - 6,800 = 4,800

4,800 represents 3/7 of what was lost for 7 hours.

we then divide 4,800 by 3 which gets 1,600.

because we know that 1,600 represents 1/7 of how much water was lost, we just have to multiply by 7, which gets 11,200.   after adding 11,200 to how much water was left at 7: 00 p.m. which was 6,800. 11,200 + 6,800 = 18,000

this means that the pool originally had 18,000 gallons in the pool.

you can also check your work by doing 18,000 - 4(1600), which becomes           1800 - 6400 which gets you 11,600.

b. 1.25 hr

step-by-step explanation:

if you have a constant speed of 60 mph, then you have a ratio between 60 mi. to 1 hr. dividing both sides by 60 we get the ratio

1 mi. to

then mult. both sides by 75 we get the ratio

75 mi. to

simplifying we get

75 mi. to

or 1.25 hr.