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score on last try: 0 of 5 pts. see details for more. > next question get a similar question you can retry this question below outside temperature over a day can be modeled as a sinusoidal function. suppose you know the temperature is 75 degrees at midnight and the high and low temperature during the day are 83 and 67 degrees, respectively. assuming t is the number of hours since midnight, find an equation for the temperature, d, in terms of t. d(t)=75 + 8 cos(π/12 t) × question help: video submit question

score on last try: 0 of 5 pts. see details for more. > next question get a similar question you can retry this question below outside temperature over a day can be modeled as a sinusoidal function. suppose you know the temperature is 75 degrees at midnight and the high and low temperature during the day are 83 and 67 degrees, respectively. assuming t is the number of hours since midnight, find an equation for the temperature, d, in terms of t. d(t)=75 + 8 cos(π/12 t) × question help: video submit question

Answer

Explanation:

Step1: Find the amplitude

The amplitude $A$ is half the difference between the high and low temperatures. $A=\frac{83 - 67}{2}=\frac{16}{2}=8$.

Step2: Find the vertical - shift

The vertical - shift $k$ is the average of the high and low temperatures. $k=\frac{83 + 67}{2}=\frac{150}{2}=75$.

Step3: Find the period and angular frequency

The period $T$ of a daily temperature cycle is 24 hours. The formula for the angular frequency $\omega$ is $\omega=\frac{2\pi}{T}$. Since $T = 24$, then $\omega=\frac{2\pi}{24}=\frac{\pi}{12}$.

Step4: Write the sinusoidal function

The general form of a cosine - based sinusoidal function is $D(t)=k + A\cos(\omega t)$. Substituting $k = 75$, $A = 8$, and $\omega=\frac{\pi}{12}$, we get $D(t)=75 + 8\cos(\frac{\pi}{12}t)$.

Answer:

$D(t)=75 + 8\cos(\frac{\pi}{12}t)$