Trigonometry Word Problem

Question

The water at a boat dock is 7 feet deep at low tide, and 11 feet deep at high tide. One day, low tide occurred at 4 am and high tide at 10 am. Find a function for the height of the tide as a function of time t .

Think about the characteristics: amplitude, period, midline, and try to sketch a figure reflecting how the tide height moves up and down.

Answer 1

Tide comes in twice a day and its 6 hours between highs and lows

Thus using a sinusoidal model, the Period T = 12 hours

The Amplitude = 1/2 Peak to Peak = 1/2 ( 11 - 7 ) = 2

The midline is at 1/2 (11 + 7) = 9 feet

The sine is zero at 9 ft so it = 0 at midpoints between 4 and 10 which is t = 13 and 7

OR = 1 and 7 if the period T = 12 hours

Using a sine function, zero would be at midpoint between 4 and 10 = 6pm/am

sin ( wt + phi ) w = 2 pi f = 2 pi / T = pi / 6

For Phase shift ( pi/6 )t + phi = 0 at t =1 and 7 If t=7 => phi = - 7pi/6

So the function would be

******* H( t ) = 2 sin ( pi/6 (t) - 7pi/6 ) + 9 ********

To check H = 11 at t = 10

H(10) = 2 sin ( pi/6 (10) - 7 pi/6 ) + 9

= 2 sin (3 pi/6) + 9

= 2 (1) + 9 = 11 check ok

And ... H = 7 at t = 4

H(4) = 2 sin ( pi/6 (4) - 7 pi/6 ) + 9

= 2 sin ( - 3 pi/6 ) +9 = 7 check ok