Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature is 55 degrees at midnight and the high and low temperature during the day are 64 and 46 degrees, respectively. Assuming \( t \) is the number of hours since midnight, find an equation for the temperature, \( D \), in terms of \( t \). \( D(t)=\square \)

To model the temperature \( D(t) \) as a sinusoidal function, we can use the general form of a sinusoidal function: \[ D(t) = A \sin(B(t - C)) + D \] where: - \( A \) is the amplitude, - \( B \) affects the period, - \( C \) is the horizontal shift (phase shift), - \( D \) is the vertical shift (midline). ### Step 1: Determine the amplitude \( A \) The amplitude \( A \) is half the difference between the maximum and minimum temperatures: \[ A = \frac{\text{max} - \text{min}}{2} = \frac{64 - 46}{2} = \frac{18}{2} = 9 \] ### Step 2: Determine the midline \( D \) The midline \( D \) is the average of the maximum and minimum temperatures: \[ D = \frac{\text{max} + \text{min}}{2} = \frac{64 + 46}{2} = \frac{110}{2} = 55 \] ### Step 3: Determine the period and \( B \) The period of the temperature cycle is 24 hours (since it repeats daily). The value of \( B \) is calculated using the formula: \[ B = \frac{2\pi}{\text{period}} = \frac{2\pi}{24} = \frac{\pi}{12} \] ### Step 4: Determine the phase shift \( C \) Since the temperature is 55 degrees at midnight (which is \( t = 0 \)), and this is also the midline, we can use a cosine function (which starts at its maximum) to model the temperature. However, since we want the temperature to start at the midline, we can use the sine function shifted appropriately. To find the phase shift, we note that the maximum temperature occurs at some point after midnight. The maximum temperature (64 degrees) occurs halfway through the cycle, which is 12 hours after midnight. Therefore, we can set \( C = 12 \). ### Step 5: Write the equation Putting it all together, we can write the temperature function as: \[ D(t) = 9 \cos\left(\frac{\pi}{12}(t - 12)\right) + 55 \] This can also be expressed using the sine function: \[ D(t) = 9 \sin\left(\frac{\pi}{12}t - \frac{\pi}{2}\right) + 55 \] However, since we are starting from the midline, the cosine form is more straightforward. ### Final Equation Thus, the equation for the temperature \( D(t) \) in terms of \( t \) is: \[ D(t) = 9 \cos\left(\frac{\pi}{12}(t - 12)\right) + 55 \] or equivalently, \[ D(t) = 9 \sin\left(\frac{\pi}{12}t - \frac{\pi}{2}\right) + 55 \] You can choose either form based on your preference.