The water level varies from 12 inches at low tide to 52 inches at high tide. Low tide occurs at 9:15 am. High tide occurs at 3:30 pm. What is a cosine function that models the variation in inches above and below the water level as a function of time and hours since 9:15 am?
My answer ism 20cos(x-3.5)+32
My dogs laying on my bed
Your amplitude and vertical shift are ok
but your period is not taken care of
period:
time between 9:15 am and 3:30 pm
= 6:15 = 6.25 hrs
so the period is 12.5 hrs
2π/k = 12.5
k = 2π/12.5 = 4π/25
height = 20 cos ( (4π/25)(x + d) ) + 32
we want 3:15 am ---> x = 0
12 = 20cos( 4π/25 d) + 32
-1 = cos( 4π/25 d)
I know cos π = -1
so 4π/25 d = π
d = 25/4 =6.25
I get
height = 20 cos ( (4π/25)(x + 6.25) ) + 32
check:
when t=0 ,
height = 20cos( (4π/25)(6.25)) + 32
= 20cos π + 32 = -20+32 = 12 , good!
when t= 6.25
height = 20 cos( (4π/25)(6.25+6.25)) + 32
= 20 cos 2π + 32 = 20+32 = 52, good!
Nice dog, what breed?
Dog is more important
Thank you
bark
How did you get 4pi/25? Is it multiplying by 2? Besides, Is your answer 100% correct?
more details about the dog, please.
To model the variation in inches above and below the water level as a cosine function, we need to consider the amplitude (half the distance between the maximum and minimum values), the period (the time it takes for one complete cycle), and the phase shift (the horizontal shift of the function).
In this case, the amplitude can be found by taking half the difference between the maximum and minimum values of the water level, which is (52 - 12) / 2 = 20 inches.
To find the period of the function, we need to determine the time it takes for one complete cycle, which is the time difference between high tide and low tide. Given that low tide occurs at 9:15 am and high tide occurs at 3:30 pm, the time difference is 6 hours and 15 minutes.
However, the cosine function operates on radians, not hours. To convert the period to radians, we need to divide the time difference by 12 and multiply by 2π (since there are 2π radians in one complete cycle). Therefore, the period of the cosine function in radians is (6.25 / 12) * 2π = 1.04π radians.
Finally, to determine the phase shift of the function, we need to identify where the function starts with respect to time since 9:15 am. Based on the given information, we can see that at 9:15 am, the function is at its minimum (low tide). Thus, the phase shift is 0, which means the function doesn't have any horizontal shift.
Combining all the information, the cosine function that models the variation in inches above and below the water level as a function of time and hours since 9:15 am can be expressed as:
f(x) = A * cos(2π * (x - D) / P) + C
Where:
A = 20 (amplitude)
D = 0 (phase shift)
P = 1.04π (period)
Therefore, the correct cosine function would be:
f(x) = 20 * cos(2π(x / 1.04π))
Note: Keep in mind that this function considers time in hours since 9:15 am, where x = 0 corresponds to 9:15 am.