Question

The desert temperature, H, oscillates daily between 40◦F at 5 am and 80◦F at 5 pm. Write a possible for- mula for H in terms of ????, measured in hours from 5 am.

Answer 1

Answer:

$H(t)=-20\text{cos}(\frac{\pi}{12}t)+60$

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.

$y=A\text{cos}[B(x-C)]+D$, where,

A = Amplitude,

$\text{Period}=\frac{2\pi}{|B|}$

C = Phase shift,

D = Vertical shift.

First of all, we will find amplitude using maximum and minimum values as:

$A=\frac{\text{Maximum value}-\text{Minimum value}}{2}$

$A=\frac{80-40}{2}$

$A=\frac{40}{2}$

$A=20$

Since period is 24 hours (5 am to 5 pm), so let us find B as:

$24=\frac{2\pi}{|B|}$

$B=\frac{2\pi}{24}$

$B=\frac{\pi}{12}$

$\text{Vertical shift}=\frac{\text{Maximum value}+\text{Minimum value}}{2}$

$D=\frac{80+40}{2}=\frac{120}{2}=60$

There is no phase shift.

Since temperature is minimum when $t=0$, so we will use negative cosine as:

$H(t)=-20\text{cos}(\frac{\pi}{12}t)+60$

Therefore, our required function would be $H(t)=-20\text{cos}(\frac{\pi}{12}t)+60$.