Question 1183251
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Outside temperature over a day can be modelled as a sinusoidal function. 
Suppose you know the high temperature for the day is 90 degrees and the low temperature of 80 degrees occurs at 6 AM. 
Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.
D(t)=
How do i go about this question
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<pre>
The difference between the highest and lowest temperature is 90 - 80 = 10 degrees;  

hence, the midline is 85 degrees and the amplitude is  {{{10/2}}} = 5 degrees.



Next, if x is the time starting from 6 am (x= 0 when the temperature is lowest), then OBVIOUSLY, the temperature as a function of x is


    D(x) = {{{85 - 5*cos(2pi*(x/24))}}}      (1)


(we use the sign  " - "  to reflect cosinusoid about the midline, and we use the ratio x/24 to model the period of 24 hours).



Now, if t is the time after midnight, then x = t-6,  so the formula (1) takes the final form


    D(t) = {{{85 - 5*cos(2pi*((t-6)/24))}}}.        <U>ANSWER</U>


Or, if you like to use degrees instead of radians, then the same formula takes an EQUIVALENT form


    D(t) = {{{85 - 5*cos(15^o*(t-6))}}}.          <U>ANSWER</U>
</pre>

Solved.


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How tutor &nbsp;@Theo solved the problem, &nbsp;is a formal presentation; &nbsp;but as far as I know, &nbsp;in reality, 

nobody solves such problems in this way, &nbsp;and &nbsp;(in my opinion) &nbsp;it is not a good way to teach students.



I am 100% convinced that after such teaching, &nbsp;although formally it is right, &nbsp;the students' understanding will be &nbsp;ZERO.



What I showed you in my post, &nbsp;is a way to solve such problems &nbsp;using  common sense 

(combined with the knowledge of elementary &nbsp;Trigonometry). &nbsp;I hope, it is much more educative.