Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature for the day is 70 degrees and the low temperature of 40 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.
To find an equation for the temperature, D, in terms of t, we can use the sine function to model the sinusoidal behavior of the temperature throughout the day.
First, let's analyze the information given. We know that the high temperature for the day is 70 degrees, which will occur at some point during the day. We're also given that the low temperature of 40 degrees occurs at 3 AM, which is 3 hours after midnight.
Let's break down what we know:
- The high temperature of 70 degrees occurs at some time during the day.
- The low temperature of 40 degrees occurs 3 hours after midnight, which is equivalent to t = 3.
Knowing this, we can construct a sinusoidal function in the form:
D = A * sin(B(t - C)) + D
Where:
- D is the temperature,
- A is the amplitude (half the difference between the high and low temperatures),
- B is the frequency (2π divided by the number of hours it takes for one complete cycle),
- C is the phase shift (the time at which D is at its lowest point),
- D is the vertical shift (the average of the high and low temperatures).
In this case, the amplitude is (70 - 40) / 2 = 15 degrees, as half the difference between the high and low temperatures is 15.
The frequency, B, can be determined by calculating the number of hours it takes for one complete cycle. Since a day is 24 hours long and completes one cycle, B can be calculated as B = 2π / 24 = π / 12.
The phase shift, C, is the number of hours between midnight and the low temperature point. In this case, C = 3.
Lastly, the vertical shift, D, is the average of the high and low temperatures, which is (70 + 40) / 2 = 55 degrees.
Putting it all together, the equation for the temperature, D, in terms of t, becomes:
D = 15 * sin((π / 12)(t - 3)) + 55