Answer:
The maximum temperatures of the reaction is 15 degree.
The minimum temperatures of the reaction is -45 degree.
The entire cycle take 6 hours.
Step-by-step explanation:
Given : The function [tex]f(t) = 30 \sin (\frac{\pi}{3}t) -15[/tex] models the temperature of a periodic chemical reaction where t represents time in hours.
To find : What are the maximum and minimum temperatures of the reaction, and how long does the entire cycle take?
Solution :
We know that, sin function lies between -1 to 1.
So, The maximum and minimum points of sin x is 1 and -1 respectively.
Now, For maximum point
Substitute [tex]\sin (\frac{\pi}{3}t)=1[/tex]
[tex]f(t) = 30(1) -15[/tex]
[tex]f(t) = 15[/tex]
Now, For minimum point
Substitute [tex]\sin (\frac{\pi}{3}t)=-1[/tex]
[tex]f(t) = 30(-1) -15[/tex]
[tex]f(t) = -45[/tex]
The maximum temperatures of the reaction is 15 degree.
The minimum temperatures of the reaction is -45 degree.
General form of sin function is [tex]y=A sin(Bx)+C[/tex]
Where,
[tex]B=\frac{2\pi}{\text{Period}}[/tex]
Comparing with given function, [tex]B=\frac{\pi}{3}[/tex]
[tex]\frac{\pi}{3}=\frac{2\pi}{\text{Period}}[/tex]
[tex]\text{Period}=\frac{2\pi\times 3}{\pi}[/tex]
[tex]\text{Period}=6[/tex]
Therefore, The entire cycle take 6 hours.
