Lighthouse B is 9 miles west of lighthouse A. A boat leaves A and sails 4 miles. At this​ time, it is sighted from B. If the bearing of the boat from B is N64degrees​E, how far from B is the​ boat?

Question

The boat is either 8.7 miles or ___ miles from lighthouse​ B, to the nearest tenth of a mile.

I don't understand how to get the second value

Answer 1

B's bearing when it sees A after it goes 4 miles West is N26E

Use Law of Sines first, to get C=80.23 or 99.77

CosC = +/- .17

solve for a using Law of Cosines

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

81 = a^2 +16-2(4)a(.17)

use quadratic formula

a= (+/-1.358+16.182)/2

at that time it's 8.76 or 7.41 miles from B

Answer 2

The angle between AB and the sighting is 26º

Label ships location C

Using Law of Cosines

4² = BC² + 9² - 2(BC)(9) cos 26º

BC = 8.74621

BC = 7.43179

Answer 3

Let's find an angle associated with the side 9:

sin(90-64)/4 = sin(x)/9

Solving, we find x ≈ 80.5171918082°.

Let's round this to 80.52°.

90° - 80.52° = 9.48°.

If another solution exists, it will be symmetric to a right angle:

90 + 9.48° = 99.48° (the other solution angle).

180° -26° - 99.48° = 54.52°.

Now, use this angle to find the second solution side:

sin(54.52°)/X = sin(26°)/4

X = sin(54.52°)*4/sin(26°) ≈ 7.43 miles

Let's check the other value:

180° - 26° - 80.52° = 73.48°

sin(73.48°)/X = sin(26°)/4

X = sin(73.48°)*4/sin(26°) ≈ 8.748 miles