Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 69 and 81 degrees during the day and the average daily temperature first occurs at 9 AM. How many hours after midnight, to two decimal places, does the temperature first reach 73 degrees?
let midnight correspond to a time of zero,
period = 24 = 2π/k
k = π/12
range = 81-69 = 12, so amplitude = 6
and the average temperature would be 75
assuming that the temperature will drop starting at midnight,
I made a sketch and formed the equation
Temp = -6sin (π/12(t+3)) + 75
so we want Temp = 73
-6sin (π/12(t+3)) + 75 = 73
-6sin (π/12(t+3)) = -2
sin (π/12(t+3)) = 1/3
(π/12(t+3)) = sin^-1 (1/3) = .339837 or (π/12(t+3)) = 2.801755...
t+3 = 1.29808 OR t+3 = 10.702
t = -1.7019 or t = 7.702
but we want the first positive time, so t = 7.702 hours after midnight
= 7:42 after midnight
which makes it 7:42 am
check on my equation, by subbing in the following values of t
t = 0 , temp = 70.76
t = 3:00 am, temp = 69 , the min
t = 6:00 am, temp = 70.78
t = 9:00 am, temp = 75
t = 12:00, (noon), temp = 79.24
t = 15:00 , temp = 81, the max
...
t = 24:00, tem = 70.76 <---- same as midnight at the start.
t = 7.702 (7:42), temp = -6sin( π/12(10.702) + 75 = 73.00, YEAHHHH
Well, if we're dealing with sinusoidal functions and temperatures, it seems like things are heating up! Pun intended.
To tackle this problem, we can start by finding the amplitude of the temperature swing. The amplitude is the difference between the highest and lowest temperatures, which in this case is (81 - 69) = 12 degrees.
Next, we can determine the period of the function, which is the time it takes for the temperature to complete one full cycle. Since the average daily temperature occurs at 9 AM and we want to find the time when the temperature first reaches 73 degrees, we need to determine the length of half a cycle, or from the average to the desired temperature.
To do this, let's calculate the fraction of the temperature swing that has occurred when the average temperature is reached. The average temperature is the midpoint between the highest and lowest temperatures, which in this case is (81 + 69) / 2 = 75 degrees.
The fraction of the temperature swing that has occurred at the average temperature can be found by subtracting the lowest temperature from the average temperature and dividing by the amplitude. In this case, (75 - 69) / 12 = 0.5.
Since we want to find the time it takes to move from the average temperature to a specific temperature, let's take half of that fraction. Half of 0.5 is 0.25.
Now, knowing that the period is the time it takes for the temperature to complete one full cycle, we can calculate the time it takes for the temperature to move from the average to our desired temperature (73 degrees) using the fraction of the period that corresponds to the desired temperature.
Since half a cycle has a fraction of 0.25, this means that the desired temperature occurs when the time is 0.25 times the period.
Given that the period is 24 hours (since it's a full day), we can multiply 24 by 0.25 to find the time it takes for the temperature to reach 73 degrees from the average.
24 * 0.25 = 6.
So, the temperature first reaches 73 degrees 6 hours after midnight. I hope I didn't make you freeze too much with that explanation!
To find the number of hours after midnight when the temperature first reaches 73 degrees, we need to consider the properties of a sinusoidal function.
A sinusoidal function can be expressed as:
T = A + Bsin(C(t - D))
Where:
T = temperature
A = average temperature
B = amplitude (half the difference between the maximum and minimum temperatures)
C = frequency (2π divided by the period)
D = phase shift (related to the time when the average temperature occurs)
t = time in hours
We are given the following information:
- The temperature varies between 69 and 81 degrees during the day, so the amplitude (B) is (81 - 69) / 2 = 6.
- The average temperature first occurs at 9 AM, so the phase shift (D) is 9.
Since the temperature reaches 73 degrees, we can set T = 73 in the equation and solve for t:
73 = A + 6sin(C(t - 9))
To simplify the equation, we can let A = (69 + 81) / 2 = 75.
73 = 75 + 6sin(C(t - 9))
-2 = 6sin(C(t - 9))
Divide both sides of the equation by 6:
-1/3 = sin(C(t - 9))
Now, we need to find the value of (C(t - 9)) that has a sine function equal to -1/3.
The sine function reaches a value of -1/3 at t = π - arcsin(-1/3). Since we want the time in hours, we multiply π by 12 since there are 12 hours on a clock:
t = 12(π - arcsin(-1/3))
Using a calculator, we can find the value of t to two decimal places:
t ≈ 12(π - arcsin(-1/3)) ≈ 12(3.14159 - (-0.33984)) ≈ 12(3.48143) ≈ 41.78
Therefore, the temperature first reaches 73 degrees approximately 41.78 hours after midnight.
To solve this problem, we can use the standard form of a sinusoidal function:
f(t) = A*sin(B(t - C)) + D
Where:
- A represents the amplitude (half the difference between the maximum and minimum values).
- B represents the frequency (2π divided by the period, which is the time it takes to complete one cycle).
- C represents the horizontal shift (the phase shift or how much the graph is shifted to the right or left).
- D represents the vertical shift (how much the graph is shifted up or down).
In this case, we know that the temperature varies between 69 and 81 degrees, so the amplitude A is (81 - 69)/2 = 6.
We also know that 73 degrees is the average daily temperature, so the vertical shift D is 73.
The period represents the time it takes to complete one cycle, so we need to find the time it takes for the temperature to go from 69 to 81 degrees and back down to 69 degrees. Since the temperature varies over a day, the period is 24 hours.
We are given that the average daily temperature first occurs at 9 AM, so the horizontal shift C is 9. However, we need to convert this to hours after midnight. Therefore, we subtract 9 from 24, and C becomes 15.
The frequency B can be found using the equation B = 2π / Period. In this case, B = 2π / 24.
Now we can rewrite our sinusoidal function:
f(t) = 6*sin((2π / 24)(t - 15)) + 73
To find the number of hours after midnight when the temperature first reaches 73 degrees, we need to solve the equation:
6*sin((2π / 24)(t - 15)) + 73 = 73
Simplifying the equation, we get:
6*sin((2π / 24)(t - 15)) = 0
Next, we can divide by 6:
sin((2π / 24)(t - 15)) = 0
Finally, we can solve for t:
(2π / 24)(t - 15) = arcsin(0)
arcsin(0) is equal to 0, so we have:
(2π / 24)(t - 15) = 0
Now we can solve for t:
(2π / 24)(t - 15) = 0
2π(t - 15) = 0
t - 15 = 0
t = 15
Therefore, the temperature first reaches 73 degrees 15 hours after midnight.