Question

A ship is anchored off a long straight shoreline that runs north and south. From two observation points 17 miles apart on shore, the bearings of the ship are N 32° E and S 53° E. What is the shortest distance from the ship to the shore? (Round your answer to the nearest tenth.)

Please explain to me how to solve this problem. Thank you!

Answer 1

From the attached figure.
1. The distance between the two observers A and B is 17.
2. shortest distance to shore is Z
3. x + y = 17 => x = 17 - y
4. z/y = tan(32) => z = y*tan(32)
5. z/x = tan(53) => z = x*tan(53)

combine step 4 and 5 to get:

6. y*tan(32) = x*tan(53)
from step 3, x = 17 - y
combine step 6 and step 3 to get:
7. y*tan(32) = (17 - y)*tan(53)
=> y= [17 tan(53)]/[tan(32) + tan(53)]

combine step 4 and step 7 to get:
8. Z={17*tan(53) *tan(32)}/[tan(32) + tan[53)]
= 7.22209205254024310884

Answer 2

Oh! That makes a lot of sense. Thank you so much for taking the time out to help me. I appreciate it!