Question

The water level varies from 12 inches at low tide to 52 inches at high tide. Low tide occurs at 9:15 a.m. and high tide occurs at 3:30 p.m. What is a cosine function that models the variation in inches above and below the water level as a function of time in hours since 9:15 a.m.?

Answer 1

from the data:
difference between high tide and low tide
= 15:30 - 9:15 = 3:15
so the period = 6:30 or 6.5 hours
the bot equation would make it as 6 hours

period = 2π / k
6.5 = 2π/k
k = 2π/6.5 = 4π/13

so the equation would be:
y = 20 cos (4π/13 t) + 32

graph this using:
desmos.co m/calculator , to see that it is correct

(of course after cutting and pasting <b>desmos.co m/calculator</b>
you know you have to delete the space to fool this webpage)

Answer 2

We can use the cosine function to model the variation in inches above and below the water level as a function of time. The general form of a cosine function is:

y = A cos(Bx - C) + D

where:

A = amplitude (half the distance between the highest and lowest points of the function)
B = period (the time it takes for one complete cycle of the function)
C = horizontal shift (the phase shift of the function)
D = vertical shift (the average or midpoint of the function)

In this case, we know that the water level varies from 12 inches at low tide to 52 inches at high tide. Therefore, the amplitude is:

A = (52 - 12) / 2 = 20 inches

We also know that the period of the function is 6 hours, since there are approximately 6 hours between low tide and high tide. Therefore:

B = 2π / 6 = π/3

The horizontal shift is the time difference between the start of the function (9:15 a.m.) and the beginning of the cycle (low tide). Since low tide occurs at 9:15 a.m., there is no horizontal shift:

C = 0

Finally, the vertical shift is the average of the highest and lowest points of the function:

D = (52 + 12) / 2 = 32 inches

Putting this all together, we get the cosine function:

y = 20 cos(π/3 x) + 32

where x is the time in hours since 9:15 a.m. To find the water level at a specific time, simply plug in the value of x into the function and evaluate. For example, to find the water level at 1:30 p.m. (4.25 hours after low tide), we have:

y = 20 cos(π/3 x) + 32
y = 20 cos(π/3 x) + 32
y = 20 cos(π/3 * 4.25) + 32
y ≈ 41.3 inches

So the water level is approximately 41.3 inches above sea level at 1:30 p.m.

Answer 3

Yes, you are correct that the period is 6.5 hours, not 6 hours as initially assumed. Therefore, the value of k in the cosine function should be 2π/6.5, which simplifies to 4π/13.

The correct equation for the water level as a function of time in hours since 9:15 a.m. is:

y = 20 cos (4π/13 t) + 32

Thank you for catching that mistake, and using Desmos.com/calculator is a great way to visually confirm the equation and its behavior.