# Application of Cosine Problem

• Feb 28th 2009, 02:31 PM
chrozer
Application of Cosine Problem
The tide, or depth of the ocean near the shore, changes throughout the day. The water depth $d$ (in feet) of a bay can be modeled by

$d = 35-28\cos{\frac {\pi}{6.2}}{t}$

where $t$ is the time in hours, with $t=0$ corresponding to 12:00 A.M.

(a) Algebraically find the times at which the high and low tides occur.
How would I do this? Just solve for $d$ ? Wouldn't I just get one answer? I need two, one for the high and one for the low.

(b) Algebraically find the time(s) at which the water depth is 3.5 feet.
Just $d$ for 3.5 and solve right? I have one question though...how would you determine $\cos {\frac {\pi}{6.2}}$ algebraically?

I have also attached a picture of the original problem.
• Feb 28th 2009, 03:04 PM
skeeter
Quote:

Originally Posted by chrozer
The tide, or depth of the ocean near the shore, changes throughout the day. The water depth $d$ (in feet) of a bay can be modeled by

$d = 35-28\cos{\frac {\pi}{6.2}}{t}$

where $t$ is the time in hours, with $t=0$ corresponding to 12:00 A.M.

(a) Algebraically find the times at which the high and low tides occur.
How would I do this? Just solve for $d$ ? Wouldn't I just get one answer? I need two, one for the high and one for the low.

(b) Algebraically find the time(s) at which the water depth is 3.5 feet.
Just $d$ for 3.5 and solve right? I have one question though...how would you determine $\cos {\frac {\pi}{6.2}}$ algebraically?

I have also attached a picture of the original problem.

$-1 \leq \cos(anything) \leq 1$

low tide (d = 7 ft) occurs when $\cos\left(\frac{\pi t}{6.2}\right) = 1$

first low tide is when $\frac{\pi t}{6.2} = 0$

high tide (d = 63 ft) occurs when $\cos\left(\frac{\pi t}{6.2}\right) = -1$

first high tide is when $\frac{\pi t}{6.2} = \pi$

part (b) has no solution ... the lowest depth is 7 ft.
• Feb 28th 2009, 03:55 PM
chrozer
Quote:

Originally Posted by skeeter
$-1 \leq \cos(anything) \leq 1$

low tide (d = 7 ft) occurs when $\cos\left(\frac{\pi t}{6.2}\right) = 1$

first low tide is when $\frac{\pi t}{6.2} = 0$

high tide (d = 63 ft) occurs when $\cos\left(\frac{\pi t}{6.2}\right) = -1$

first high tide is when $\frac{\pi t}{6.2} = \pi$

part (b) has no solution ... the lowest depth is 7 ft.

Ok so I found that the solutions to the low tides is at when t = 0, 12.4, 24.8, and so on.
So the times are 12:00 A.M., 12:24 P.M. and 12:48 A.M. So low tides occur every 12 hours and 24 minutes from 12:00 A.M. right?

And the solutions to the high tide is at when t = 6.2, 18.6, 31, and so on.
So the times are 6:12 A.M., 6:36 P.M. and 7:00 A.M. So high tides occur every 12 hours and 24 minutes from 6:12 A.M. right?
• Feb 28th 2009, 05:26 PM
skeeter
correct ...

period of the cycle, $T = \frac{2\pi}{\frac{\pi}{6.2}} = 12.4$ hrs