Related Rates (Calculus)

Question

A lighthouse is on an island 250 meters off shore. The lighthouse beam makes a full turn every 15 seconds. How rapidly is the point where the beam meets the shoreline moving along the shoreline when the beam meets the shoreline at a point 1200 meters from the lighthouse?

Answer 1

a right triangle with hypotenuse 1200 meters and one side 250 meters has the 3rd side = sqr(1200^2-250^2) = sqr(1440000- 62500) = about 1173.7 meters

h^2 = a^2 + b^2 (hypotenuse squared = sum of the other two sides squared)

differentiate with respect to time

2hh' = 2aa' + 2bb' but b'=0 b=250, h=1200, a=1200sin78 = .98(1200) = 1173.7

hh' = aa'

h' =aa'/h = 1173.7a'/1200 = about 0.98a'

a = h(a/h) = hsinx x is the angle of the right triangle between sides 250 and 1200

a' = hcosx(x') + sinx(h') = 1200(.21)2pi/15 +.98(.98a') the angle changes at the rate of 2pi/15 seconds

a'- .96a = .04a = 33pi

a' = 33pi/.04 = 825pi meters per second = rate of change of the beam moving along the shoreline when it's 1200 meters from the shore line to the lighthouse

(a lot of rounding approximations)