Chapter 12.8, Problem 34PPS

To determine

Find an equation that represent the height h of the water t hours after noon on the first day.

Answer to Problem 34PPS

The equation for the sine function will be $h(t)=6\sin \frac{\pi }{9}(t-4.5)+9$ .

Explanation of Solution

Given: It is given in the question that the height of the water in a harbor rose to a maximum height of $15ft$ at $6:00pm$ and then dropped to a maximum level of $3ft$ by $3:00am$ . The water level can be modeled by the sine function.

Concept Used:

In this, use the concept of the general equation for sine $h=a\sin b(t-c)+d$ , where a is amplitude, $\frac{2\pi }{\left|b\right|}$ is period and d is the vertical shift . The midline will be the average of the maximum and minimum value and the amplitude is the difference between midline and maximum value.

Calculation: Here, the maximum value is $15ft$ and the minimum value is $3ft$ .

Since, it has to find the midline value,

  $d=\frac{15+3}{2}=\frac{18}{2}=9$

The vertical shift is $d=9$ .

Now, the amplitude is the difference between midline and maximum value:

  $\begin{array}{l}15-9=6\\ a=6\end{array}$

Since, it takes nine hours from maximum which is a half of the period.

Now, it is known that the period is $18$ .

  $\begin{array}{l}18=\frac{2\pi }{b}\\ b=\frac{2\pi }{18}\\ b=\frac{\pi }{9}\end{array}$

For the function $y=\sin x$ one of the points on the midline is $(0,0)$ .

In this case, one of the points on the midline is $(4.5,9)$ .

It is known that x − coordinate is $4.5$ midline is on the half of the way between minimum and maximum and it takes $9$ hours from minimum to maximum so $\frac{9}{2}=4.5$ .

It means that the phase shift is $4.5$ units to the right, so $c=4.5$ .

Now, put all the values a, b, c, d on the equation:

  $h(t)=6\sin \frac{\pi }{9}(t-4.5)+9$

Conclusion:

The equation is $h(t)=6\sin \frac{\pi }{9}(t-4.5)+9$ .