Respuesta :
Answer:
[tex]H(t)=-20\text{cos}(\frac{\pi}{12}t)+60[/tex]
Step-by-step explanation:
We have been given that the desert temperature, H, oscillates daily between 40◦F at 5 am and 80◦F at 5 pm. We are asked to write a formula H in terms of t, measured in hours from 5 am.
We will use cosine function to write our required formula.
[tex]y=A\text{cos}[B(x-C)]+D[/tex], where,
A = Amplitude,
[tex]\text{Period}=\frac{2\pi}{|B|}[/tex]
C = Phase shift,
D = Vertical shift.
First of all, we will find amplitude using maximum and minimum values as:
[tex]A=\frac{\text{Maximum value}-\text{Minimum value}}{2}[/tex]
[tex]A=\frac{80-40}{2}[/tex]
[tex]A=\frac{40}{2}[/tex]
[tex]A=20[/tex]
Since period is 24 hours (5 am to 5 pm), so let us find B as:
[tex]24=\frac{2\pi}{|B|}[/tex]
[tex]B=\frac{2\pi}{24}[/tex]
[tex]B=\frac{\pi}{12}[/tex]
[tex]\text{Vertical shift}=\frac{\text{Maximum value}+\text{Minimum value}}{2}[/tex]
[tex]D=\frac{80+40}{2}=\frac{120}{2}=60[/tex]
There is no phase shift.
Since temperature is minimum when [tex]t=0[/tex], so we will use negative cosine as:
[tex]H(t)=-20\text{cos}(\frac{\pi}{12}t)+60[/tex]
Therefore, our required function would be [tex]H(t)=-20\text{cos}(\frac{\pi}{12}t)+60[/tex].
