Chapter 6.6, Problem 34AYU

Tides The length of time between consecutive high tides is 12 hours and 25 minutes. According to the National Oceanic and Atmospheric Administration, on Saturday, March 28. 2015, in Sitka, Alaska, high tide occurred at $8:06$ AM ( $8.10$ hours) and low tide occurred at $2:48$ PM ( $14.8$ hours). Water heights arc measured as the amounts above or below' the mean lower low water. The height of the water at high tide was $8.13$ feet, and the height of the water at low tide was $1.95$ feet.

(a) Approximately when will the next high tide occur?

(b) Find a sinusoidal function of the form $y=A\sin (\omega x-\varphi )+B$ that models the data.

(c) Use the function found in part $\left(b\right)$ to predict the height of the water at 6 PM on March 28, 2015.

To determine

To find:

a. Approximately when will the next high tide occur?

Answer to Problem 34AYU

Solution:

a. 8:31 PM

Explanation of Solution

Given:

The length of time between consecutive high tides is 12 hours and 25 minutes. According to the National Oceanic and Atmospheric Administration, on Saturday, March 28, 2015, in Sitka, Alaska, high tide occurred at 8:06 am (8.10 hours) and low tide occurred at 2:48 pm (14.8 hours). Water heights are measured as the amounts above or below the mean lower low water. The height of the water at high tide was 8.13 feet, and the height of the water at low tide was 1.95 feet.

Calculation:

a. Approximately when will the next high tide occur?

Charleston, South Carolina, high tide occurred at 8:06 am, since the length of time between consecutive high tides is 12 hours and 25 minutes. Therefore the next high tide occur $=8:06AM+12:25=8:31PM$.

To determine

To find:

b. Find a sinusoidal function of the form $y=A\sin \left(\omega x-\varphi \right)+B$ that models the data.

Answer to Problem 34AYU

Solution:

b. $y=3.09\sin \left(\frac{24\pi }{149}x+0.4575\right)+5.04$

Explanation of Solution

Given:

The length of time between consecutive high tides is 12 hours and 25 minutes. According to the National Oceanic and Atmospheric Administration, on Saturday, March 28, 2015, in Sitka, Alaska, high tide occurred at 8:06 am (8.10 hours) and low tide occurred at 2:48 pm (14.8 hours). Water heights are measured as the amounts above or below the mean lower low water. The height of the water at high tide was 8.13 feet, and the height of the water at low tide was 1.95 feet.

Calculation:

b. Find a sinusoidal function of the form $y=A\sin \left(\omega x-\varphi \right)+B$ that models the data.

Determine A, the amplitude of the function.

$\text{Amplitude }=\frac{ \text{largest data value }-\text{ smallest data value} }{2}=\frac{8.13-1.95}{2}=\frac{6.18}{2}=3.09$

Determine B, the vertical shift of the function.

$\text{Vertical shift }=\frac{ \text{largest data value }+\text{ smallest data value}}{2}=\frac{8.13+1.95}{2}=\frac{10.08}{2}=5.04$

Determine $\omega$, It is easier to find the horizontal stretch factor first. Since the period of the function is $T=12.41$. Therefore $\omega =\frac{24\pi }{149}$.

Now superimpose the graph of $y=3.09\sin \left(\frac{24\pi }{149}x\right)+5.04$

Determine the horizontal shift of the function by using the period of the data.

$0.9042$ units to the right by replacing $x$ by $x-0.9042$

$y=3.09\sin \left(\frac{24\pi }{149}\left(x-0.9042\right)\right)+5.04$

$y=3.09\sin \left(\frac{24\pi }{149}x+0.4575\right)+5.04$

To determine

To find:

c. Use the function found in part (b) to predict the height of the water at 6 pm on March 28, 2015.

Answer to Problem 34AYU

Solution:

c. $3.97$

Explanation of Solution

Given:

The length of time between consecutive high tides is 12 hours and 25 minutes. According to the National Oceanic and Atmospheric Administration, on Saturday, March 28, 2015, in Sitka, Alaska, high tide occurred at 8:06 am (8.10 hours) and low tide occurred at 2:48 pm (14.8 hours). Water heights are measured as the amounts above or below the mean lower low water. The height of the water at high tide was 8.13 feet, and the height of the water at low tide was 1.95 feet.

Calculation:

c. Use the function found in part (b) to predict the height of the water at 6 pm on March 28, 2015.

$y=3.09\sin \left(\frac{24\pi }{149}\left(6\right)+0.4575\right)+5.04$

$y=3.09\times -0.3448+5.04$

$y=3.97\text{ ft}$.