Answer:
A. The next high tide occurs at 6:42 P.M.
B. The period T is 12.4 hs.
C. The amplitude of the oscillation is 3.675 feet.
D. The function is of the form [tex]Tide(t)=X_{0}+Asin(\omega t+\varphi)[/tex], with t in hours. This function is:
[tex]Tide(t)=3.375feet+(3.675 feet)sin(0.5067\frac{1}{hs} t-1.621rad)[/tex]
E. The tide will reach 6 feet for first time at 4:16 A.M.
Explanation:
A. The high tide will repeat itself after a full period, which is 12:24 hours. That means that the next high tide will occur at t=6:18hs+12:24hs=18:42hs=6:42P.M.
B. For the periode we can use a direct Rule of Three:
60 min ↔ 1h
24 min ↔ x=24/60=0.4 ⇒ T=12.4hs
C. Amplitud and initial hight can be obtain as:
[tex]A=\frac{X_{max}-X_{min}}{2}=\frac{7.05feet-(-0.3feet)}{2}=3.675feet[/tex]
[tex]X_{0}=A+X_{min}=3.675feet-0.3feet=3.375feet[/tex]
D. As said above, the function is of the form [tex]Tide(t)=X_{0}+Asin(\omega t+\varphi)[/tex], with t in hours. With this form:
[tex]\omega=\frac{2\pi}{T}= \frac{2\pi}{12.4hs}=\frac{0.5067rad}{hs}[/tex]
The maximum tide happens when the sin(X)=1 ↔ αmax=π/2.
[tex]\alpha_{max}=\frac{\pi}{2} Rad=\omega t_{max}+\varphi\\\varphi=\frac{\pi}{2} Rad-\frac{0.5067rad}{hs}6.3hs=-1.621rad[/tex]
E. To know the time at wich the tide will be 6 feet, we introduce that data in the function:
[tex]Tide(t)=6feet=3.375feet+(3.675 feet)sin(0.5067\frac{1}{hs} t-1.621rad)\\\frac{5}{7}=sin(0.5067\frac{1}{hs} t-1.621rad)\\Arcsin(\frac{5}{7})+1.621rad=0.5067\frac{1}{hs}t\\t=4.77hs\approx4:16hs[/tex]
