Question

Two lighthouses are located 60 miles from one another on a north-south line. If a boat is spotted S 21o E from the northern lighthouse and N 16o E from the southern lighthouse, how much closer is the closest lighthouse to the boat than the lighthouse furthest away?

The northern lighthouse is 8.2 miles closer than the southern lighthouse.

The southern lighthouse is 27.4 miles closer than the southern lighthouse.

The northern lighthouse is 39.6 miles closer than the southern lighthouse.

The southern lighthouse is 63.1 miles closer than the southern lighthouse.

Answer 1

(A)The northern lighthouse is 8.2 miles closer than the southern lighthouse.

Explanation:

The triangle attached represents the given problem.

First, let us determine the distance of the Boat from each of the lighthouse.

In Triangle ABC,

∠A+∠B+∠C=180 degrees

21+∠B+16=180

∠B=180-37=143 degrees.

Using Law of Sines

`(a)/(Sin A)=(b)/(Sin B)\\(a)/(Sin 21^0)=(60)/(Sin 143^0) \\\text{Cross Multiply}\\a*sin143=60*sin21\\a=60*sin21/ sin143\\a=35.73 miles`

Similarly

`(c)/(Sin C)=(b)/(Sin B)\\(c)/(Sin 16^0)=(60)/(Sin 143^0) \\\text{Cross Multiply}\\c*sin143=60*sin16\\c=60*sin16/ sin143\\c=27.48 miles`

Difference in Distance =35.73-27.48=8.25 miles

Therefore, the northern lighthouse is 8.2 miles closer than the southern lighthouse.