Math Question help

Question

The nearby beach has an average depth of the water at low tide of 1 m. The average depth of the water at high tide is 8 m. One complete cycle takes 12 h.

(a) Determine the equation that models the tides using cosine as the base function.

(b) What is the depth of the water at 02:00 (2:00 am)?

(c) Many people dive from this pier during the day. If the water must be at least 3 m deep to dive safely, between what daylight hours should people dive?

Answer 1

h(x) = Acos[B(x-C)] +D

x= time of day = t

A = amplitude=(8-1)/2=3.5 meters

2pi/B=period, phase or cycle=12 hours

B=pi/6 = 30 degrees

C is phase shift, which depends on time of day for low or high tide

which depends on the moon cycle and place on earth.

Assume high tides at about noon and midnight

low at 6am and 6pm

D= midline = (8+1)/2 = 4.5 hours

h(t) = 3.5cos[(pi/6)(t-C)]+4.5

calculate C by setting h=1 for low tide at 6am

3.5cos[(pi/6)(6-C)]+4.5=1

cos[(pi/6)(6-C)]=-1

[pi/6)(6-C)]=pi

6-C =pi/(pi/6)=6

-C=0

C=0

a)

h(x)=Acos[B(x-C)]+D

plug in the values from above

h(t)=3.5cos[(pi/6)t]+4.5

b)

h(2am) = 3.5cos[(pi/6)2)]+4.5

= 3.5cos(pi/3) +4.5

=3.5(1/2)+4.5

=1.75+4.5 =6.25 meters high at 2am

c)

h(t)=3 = 3.5cos[(pi/6)t]+4.5

solve for t

3.5cos[(pi/6)t]=-1.5

cos[(pi/6)t]=-1.5/3.5=-3/7

(pi/6)t = cos^-1(-3/7)

t = [cos^-1(-3/7)]/(pi/6)

t =about 115.38/30

=about 3.646

=3 hours + 60(.656) minutes

=about 3:39pm

with high tide at noon

from 8:21am to 3:39pm, the depth=3 or more meters during daylight hours

from 8:21pm to 3:39am the depth =3 or more meters at night time hours

Answer 2