Asked by Bubba
A lighthouse is fixed 130 feet from a straight shoreline. A spotlight revolves at a rate of 11 revolutions per minute, (22 rad/min ), shining a spot along the shoreline as it spins. At what rate is the spot moving when it is along the shoreline 12 feet from the shoreline point closest to the lighthouse?
Answers
Answered by
Reiny
At a given time of t minutes, let the light be x ft down the shore
let the angle formed at the lighthouse with the perpendicular to the shore be Ø
so tanØ = x/130
x = 130tanØ
dx/dt = 130 sec^2 Ø dØ/dt
given dØ/dt = 11(2π) or 22π radian/min (you had 22)
when x = 12
when x = 12, and the other side = 130
hypot^2 = 12^2 + 130^2
hypot = √1704
cosØ = 130/√17044
sec Ø = √17044/130
sec^2 Ø = 17044/16900 = 4261/4225
then dx/dt = 130(4261/4225)(22π)
or appr. .....
you do the button-pushing.
let the angle formed at the lighthouse with the perpendicular to the shore be Ø
so tanØ = x/130
x = 130tanØ
dx/dt = 130 sec^2 Ø dØ/dt
given dØ/dt = 11(2π) or 22π radian/min (you had 22)
when x = 12
when x = 12, and the other side = 130
hypot^2 = 12^2 + 130^2
hypot = √1704
cosØ = 130/√17044
sec Ø = √17044/130
sec^2 Ø = 17044/16900 = 4261/4225
then dx/dt = 130(4261/4225)(22π)
or appr. .....
you do the button-pushing.
Answered by
H H Chau
Concur.
At x=12 ft
dx/dt=(12^2+130^2)/130 * 22π=9062 ft/min
At x=12 ft
dx/dt=(12^2+130^2)/130 * 22π=9062 ft/min