Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature for the day is 57 degrees and the low temperature of 43 degrees occurs at 3 AM. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.
Assume y = m + Asin(k(t-h))
center line at (57+43)/2 = 50
y = 50+sin(k(t-h))
amplitude is (57-43)/2 = 7
y = 50+7sin(k(t-h))
minimum is at t=3, so
y = 50 - 7cos(k(t-3))
period is 24 hours, so 2π/k = 24
y = 50 - 7cos(π/12 (t-3))
To find an equation for the temperature, D, in terms of t, let's use the following steps:
Step 1: Determine the amplitude.
The amplitude, A, is the difference between the average temperature and either the high or low temperature. In this case, the average temperature is the average of the high and low temperatures. Thus, the amplitude is (57 - 43) / 2 = 7.
Step 2: Determine the period.
The period, P, is the time it takes for the temperature to complete one full cycle. In this case, since the temperature follows a daily pattern, the period is 24 hours.
Step 3: Determine the phase shift.
The phase shift, C, represents any horizontal shift in the sinusoidal function. In this case, the low temperature occurs at 3 AM, which is 3 hours after midnight. So the phase shift is -3.
Step 4: Write the equation.
The general equation for a sinusoidal function is D = A * sin(B * (t - C)) + D_avg, where B = 2π/P.
Substituting the values we found:
D = 7 * sin((2π/24) * (t - (-3))) + (57 + 43) / 2
Simplifying further:
D = 7 * sin((π/12) * (t + 3)) + 50
Therefore, the equation for the temperature, D, in terms of t is:
D = 7 * sin((π/12) * (t + 3)) + 50
To find the equation for the temperature, D, in terms of t, we need to consider the characteristics of a sinusoidal function.
First, let's analyze the given information:
- The high temperature for the day is 57 degrees.
- The low temperature of 43 degrees occurs at 3 AM.
From this, we can deduce the following:
- The amplitude of the sinusoidal function, which is half the difference between the high and low temperatures, is (57 - 43)/2 = 7 degrees.
- The midpoint of the sinusoidal function (the average of the high and low temperatures) is (57 + 43)/2 = 50 degrees.
- The period of the sinusoidal function is 24 hours, as it represents a daily cycle.
Now, let's construct the equation for the temperature, D, based on this information:
Given that the midpoint of the sinusoidal function is 50 degrees, we can start with the equation:
D = A * sin(B * t) + C,
where A is the amplitude, B is the period, and C is the midpoint.
Substituting the known values into the equation, we have:
D = 7 * sin((2π/24) * t) + 50.
Thus, the equation for the temperature, D, in terms of t is:
D = 7 * sin((π/12) * t) + 50.