outside temperature over a day can be modeled as a sinusoidal function. suppose you know the temperature is…
outside temperature over a day can be modeled as a sinusoidal function. suppose you know the temperature is 70 degrees at midnight and the high and low temperature during the day are 79 and 61 degrees, respectively. assuming t is the number of hours since midnight, find an equation for the temperature, d, in terms of t. d(t) = question help: video
Answer
Explanation:
Step1: Find the amplitude
The amplitude $A$ is half the difference between the maximum and minimum values. So $A=\frac{79 - 61}{2}=\frac{18}{2}=9$.
Step2: Find the vertical shift
The vertical shift $k$ is the average of the maximum and minimum values. So $k=\frac{79 + 61}{2}=\frac{140}{2}=70$.
Step3: Find the period
The period of a day - long cycle is $T = 24$ hours. The formula for the angular frequency $\omega=\frac{2\pi}{T}$, so $\omega=\frac{2\pi}{24}=\frac{\pi}{12}$.
Step4: Determine the phase - shift
Since the temperature is 70 degrees at midnight ($t = 0$) and the function is a sine - type function, and the mid - line value is 70, the phase - shift $h = 0$. The general form of a sinusoidal function is $D(t)=A\sin(\omega(t - h))+k$. Substituting the values of $A = 9$, $\omega=\frac{\pi}{12}$, $h = 0$, and $k = 70$ into the formula, we get $D(t)=9\sin(\frac{\pi}{12}t)+70$.
Answer:
$D(t)=9\sin(\frac{\pi}{12}t)+70$