SOLUTION: From the top of a lighthouse 75m above sea level, the angles of depression of two buoys due north of the lighthouse are 60 degrees and 30 degrees respectively. Find the exact dista

Question 1126068: From the top of a lighthouse 75m above sea level, the angles of depression of two buoys due north of the lighthouse are 60 degrees and 30 degrees respectively. Find the exact distance between the two buoys.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
in general, the angle of depression is equal to the angle of elevation.

therefore you have 2 triangles, both of which have a vertical side of 75 meters.

label the buoy with the 60 degree angle of depression C.
label the buoy with the 30 degree angle of depression D.
label the base of the lighthouse A.
label the top of the lighthouse B.

the length of AB is 75 meters.
the first buoy forms triangle ABC with the lighthouse.
the second buoy forms triangle ABD with the lighthouse.

angle ACB = 60 degrees.
anle ADB = 30 degrees.

tan(60) = AB / AC = 75 / AC.
tan(60) = sqrt(3), therefore:
sqrt(3) = 75 / AC.
solve for AC to get AC = 75 / sqrt(3) = (75 * sqrt(3)) / 3 = 25 * sqrt(3).

tan(30) = 75 / AD.
tan(30) = sqrt(3) / 3, therefore:
sqrt(3) / 3 = 75 / AD
solve for AD to get AD = 75 / (sqrt(3) / 3) = (75 * 3) / sqrt(3) = (75 * 3 * sqrt(3) / 3 = 75 * sqrt(3).

you have AC = 25 * sqrt(3) and AD = 75 * sqrt(3).

CD is equal to AD - AC = 75 * sqrt(3) - 25 * sqrt(3) = 50 * sqrt(3).

you have AC = 25 * sqrt(3) and CD = 50 * sqrt(3).

the distance between the buoys is equal to CD which is equal to 50 * sqrt(3).

my diagram is shown below:

$$$

the angle of depression is equal to the angle of elevation because they form
alternate interior angle to the parallel lines of BE and AD.

RELATED QUESTIONS
a lighthouse keeper is in the top of a lighthouse 95 feet above sea level. she notes that (answered by mananth)

the top of a lighthouse is 120 meters above sea level. the angle of depression from the... (answered by edjones)

Hi from a lighthouse 70m above sea level a ship 1.2km out to sea is observed. What is... (answered by rothauserc,MathTherapy)

From the top of a... (answered by richwmiller)

a lighthouse is in the top of a light house the angle of depression to a rock jutting... (answered by Earlsdon)

From the top of a lighthouse 50 meters high, the angle of depression of a boat out at sea (answered by scott8148)

Mia stands in a lighthouse 35 meters above sea level. She spots a whale at a 23 degree... (answered by josgarithmetic)

The observation deck at Peggy's Cove lighthouse, in Nova Scotia, is about 20 m above sea... (answered by ikleyn)