Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 67 degrees occurs at 5 PM and the average temperature for the day is 60 degrees. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.
the amplitude is 67-60, so start with
D(t) = 60 + 7 sin(t)
Since the period is 24 hours,
D(t) = 60 + 7 sin(π/12 t)
Since the high occurs at t=17, use cosine, instead of sine.
D(t) = 60 + 7 cos(π/12 (t-17))
Well, if the temperature can be modeled as a sinusoidal function, we can start by assuming it takes the form D = A*sin(Bt + C) + D0, where A represents the amplitude, B represents the frequency, C represents a phase shift, and D0 represents the average temperature.
In this case, we know that the high temperature of 67 degrees occurs at 5 PM, which is 17 hours since midnight. So, we have D(17) = A*sin(B*17 + C) + 60 = 67.
Now, we also know that the average temperature for the day is 60 degrees. This means that D0 = 60.
To find the equation, we need to determine the values of A, B, and C. Since the high temperature occurs at t = 17, we can use this information to solve for B and C.
First, let's rewrite the equation in terms of B and C: 67 = A*sin(B*17 + C) + 60.
Subtracting 60 from both sides gives us 7 = A*sin(B*17 + C).
Now, we need to find the value of sin(B*17 + C) that gives us 7. We can use the inverse sine function to solve for B*17 + C.
sin^(-1)(7/A) = B*17 + C.
Now, we have an equation with two unknowns (B and C), so we need more information to solve for them.
To find the equation for temperature, D, in terms of t, we'll use the general equation for a sinusoidal function which is:
D = A * sin(B * (t - C)) + D0
Where:
D is the temperature in degrees
A is the amplitude of the function (half the difference between the maximum and minimum values)
B is the frequency or period of the function (in this case, 2π)
C is the phase shift (t value when the maximum temperature occurs)
D0 is the vertical shift (average temperature)
Given information:
The high temperature of 67 degrees occurs at 5 PM, which is 17 hours since midnight. So C = 17.
The average temperature for the day is 60 degrees. So D0 = 60.
We also need to find A, which is half the difference between the maximum and average temperatures. The difference in this case is 67 - 60 = 7. So A = 7/2 = 3.5.
Now we can plug these values into the equation:
D = 3.5 * sin(2π * (t - 17)) + 60
Thus, the equation for the temperature D in terms of t is: D = 3.5 * sin(2π * (t - 17)) + 60.
To find an equation for the temperature, D, in terms of the number of hours since midnight, t, we can start by using the information given.
First, let's determine the amplitude of the sinusoidal function. The high temperature of 67 degrees represents the peak of the function, so the amplitude is half of the peak-to-peak temperature range. In this case, the range is 67 - 60 = 7 degrees, so the amplitude is 7 / 2 = 3.5 degrees.
Next, we need to find the period of the sinusoidal function, which represents the time it takes for the temperature to complete one full cycle. Since the high temperature of 67 degrees occurs at 5 PM, which is 5 hours after midnight, we can infer that a complete cycle occurs every 24 hours. Therefore, the period is 24 hours.
Using the general form of a sinusoidal function, D(t) = A * sin(2π / T * t) + C, where A is the amplitude, T is the period, and C is the vertical displacement or average temperature, we can plug in the values we have:
D(t) = 3.5 * sin(2π / 24 * t) + 60
This equation gives the temperature D at any given time t in hours since midnight.