A ship captain at sea uses a sextant to sight an angle of elevation of 37 degrees to the top of a lighthouse.?

Question

A ship captain at sea uses a sextant to sight an angle of elevation of 37 degrees to the top of a lighthouse.?

Answer 1

Any decent captain should be able to determine a distance like that by just looking at it. And is the base of the lighthouse at the same level as the sextant?



As a pure math problem, instead of a practical sailing problem, here is the answer.

A = point of first sighting

B = point of second sighting

T = top of lighthouse

Triangle ABT has angles 37 degrees, 130 degrees, and 13 degrees. The base is 250 feet. The length AT is 850 feet.



H = base of lighthouse

Now look at triangle AHT, it is a right triangle. Length AT is 850, as figured above. The base AH is 680 feet. The ship has moved 250 feet of this, and the rocks are at 100 feet. So the ship to rock distance is 680 - 250 -100 = 330 feet.

Answer 2

330 feet. It's difficult to explain without a diagram.



Draw a triangle with the lighthouse at the right. Let the distance between the base of the Lh and the ship be x. This gives your first equation: tan 37 = H/x



Draw a second triange with the ship 250 feet closer to the Lh. This then gives your 2nd equation: tan 50 = H/(x - 250)



Do your simultaneous equations on these to determine x, which works out at 680 feet.



The question askes distance from rocks at the 2nd sighting, which is 680 - 250 - 100 feet (because the rocks extend 100 feet) = 330 feet.

Answer 3

H = d tan 37 = (d - 250) tan 50

d tan 37 = (d - 250) tan 50

(d - 250) / d = tan 37 / tan 50

1 - 250 / d = tan 37 / tan 50

1 - tan 37 / tan 50 = 250 / d

d = 250 / { 1 - tan 37 / tan 50 } = 680

d - 250 = 430

430 - 100 = 330



Actually, this is an approximation that fails to correct for DIP, the altitude of the navigator's eye above the water as the sextant is being used.



I should say, for the benefit of MorningFoxNorth, that usually the horizon is used as the baseline for nautical altitude measurements. If the navigator were using the base of the lighthouse, instead, then the geometry of the problem changes, such that



H = sqrt(h^2 + d^2) sin q / cos[q - arctan(h/d)]



Where...

H = the vertical distance from the base of the lighthouse to its top.

h = the vertical distance of the navigator's eye above the base of the lighthouse.

d = the horizontal distance from the navigator (on the ship) to the base of the lighthouse.

q = the measured angle from the lighthouse's base to its top.



Notice that when h=0, H = d tan q