Answer:
The northern lighthouse is approximately closer to the boat than the southern lighthouse.
Step-by-step explanation:
Refer to the diagram attached. Denote the northern lighthouse as , the southern lighthouse as
, and the boat as
. These three points would form a triangle.
It is given that two of the angles of this triangle measure (northern lighthouse,
) and
(southern lighthouse
), respectively. The three angles of any triangle add up to
. Therefore, the third angle of this triangle would measure
(boat
.)
It is also given that the length between the two lighthouses (length of ) is
.
By the law of sine, the length of a side in a given triangle would be proportional to the angle opposite to that side. For example, in the triangle in this question, is opposite to side
, whereas
is opposite to side
. Therefore:
.
Substitute in the known measurements:
.
Rearrange and solve for the length of :
.
(Round to at least one more decimal places than the values in the choices.)
Likewise, with is opposite to side
, the following would also hold:
.
.
.
In other words, the distance between the northern lighthouse and the boat is approximately , whereas the distance between the southern lighthouse and the boat is approximately
. Hence the conclusion.
