Find Distance of Ship?

Question

Rita, the captain of a ship at sea, spots two lighthouses that she knows are 3 miles apart along a straight seashore. She determines that the angles between the two line-of-sight observations of the lighthouses and the line from the ship directly to shore are 15 degrees and 20 degrees.

How far is the ship from shore?

Answer 1

let x = perpendicular dist of ship to shore line
let y = distance of ship to the nearest lighthouse
tan 15 = x / ( 3 + y ) . . . . . . equation 1
tan 20 = x / y . . . . . equation 2
y = x / tan 20 . . . substitute to equation 1
tan 15 = x / (3 + x / tan 20 )
x tan 15 = 3 + x / tan 20
x = 3 / [ 1 / tan 15 - 1 / tan 20 ]
x = 3.047 miles

Answer 2

you rather could use the Pythagorean theorem in this one, because of the fact the ships are traveling at top angles to one yet another. the gap between the two ships often is the sq. root of the sum of the squares of the distances each and every deliver has taken. In time t, the 1st deliver is going 20t miles north and the 2nd deliver is going 15t miles west, so D = sqrt (400t^2 + 225t^2) = sqrt (625 t^2) = 25t which ability D=25t is the gap between the ships. i'm hoping it is clever. in case you draw a image, you could see this as a top triangle that in basic terms retains increasing because of the fact the ships head remote from the place they began.

Answer 3

Draw the sketch. Let x be the distance from shore.
You have two triangles whose adjacent sides add up to three miles.
xtan15 + xtan20 = 3
x(tan15 + tan20) = 3
x = 3/(tan15 + tan20)
x = 4.747 miles