Question

A ship is anchored off a long straight shoreline that runs north and south. From two observation points 15 mi apart on shore, the bearings of the ship are N 31 degrees E and S 53 degrees E. What is the shortest distance from the ship to the shore?

Answer 1

`color(magenta)(6.2` `"miles to the nearest 1 decimal place"`

Explanation:

`:."Let the triangle with the ship and shorline be ABC and let the shortest distance be AD"`

`:.angle ADC and angle ADB=90^@`

`:.In triangle ABD angle "BAD"= 180^@-(90^@+31^@)=59^@`

`:.In triangle ACD angle "CAD"= 180^@-(90^@+53^@)=37^@`

`:.angle BAC=59^@+37^@=96^@`

Sine rule

`:.(AB)/sinC=(BC)/sinA`

`:.AB=(BC*sinC)/sin A`

`:.AB=(15*sin 53)/sin 96^@`

`:.AB=12.046mi`

`:.AD=sin31^@*12.046`

`:.color(magenta)(=6.204mi="shortest distance"`

check:-

`:.(AC)/sinB=(AB)/sinC`

`:.(AC)/(sin31^@)=(12.046*sin31)/sin53^@`

`:.AC=7.768mi`

`:.color(magenta)(AD=cos37^@*7.768=6.204mi`