Question

A tsunami can be more than 100 feet in height and can travel at great speeds. Let a tsunami be represented by an equation of the form y= a cosbt . Suppose a wave has a height of 25 feet from its lowest depression, a period of 30 minutes, and is traveling at the rate of 180 ft/sec. How fast is the wave rising (or falling) when y= 10 feet?

I NEED EXPLNATION WITH DETAILS

Answer 1

here,

amplitude , a = 100 feet

a = 30.48 m

y = a * cos(bt)

differentiating thee equation

dy/dt = a * b * sin(bt)

dy/dt = - a*b*sqrt( 1 - cos^2(bt))

time period , T = 30 min

T = 1800 s

angular frequqncy , w = 2 * pi /T

w = 2 * pi /1800 = 3.49 * 10^-3 rad/s

dy/dt = - a*b*sqrt( 1 - (y/a)^2)

dy/dt = - (3.49 * 10^-3)*(30.48)*sqrt( 1 - (10/(30.48))^2)

dy/dt = - 0.1 m/s

the speed of fall is 0.1 m/s