Question

The function f(t) = 30 sin (pi over 3t) −15 models the temperature of a periodic chemical reaction where t represents time in hours. What are the maximum and minimum temperatures of the reaction, and how long does the entire cycle take

Answer 1

F(t)=30 sin(pi/3 t)-15
when sin(pi/3 t) = 1, f(t) = 15 => max.
when sin(pi/3 t) = -1 f(t) = -45 .+ min.
period = 2π / (π/3) = 6 hrs.

Answer 2

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 $f(t) = 30 \sin (\frac{\pi}{3}t) -15$ 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 $\sin (\frac{\pi}{3}t)=1$      

$f(t) = 30(1) -15$

$f(t) = 15$

Now, For minimum point

Substitute $\sin (\frac{\pi}{3}t)=-1$      

$f(t) = 30(-1) -15$

$f(t) = -45$

The maximum temperatures of the reaction is 15 degree.

The minimum temperatures of the reaction is -45 degree.

General form of sin function is $y=A sin(Bx)+C$

Where,

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

Comparing with given function, $B=\frac{\pi}{3}$

$\frac{\pi}{3}=\frac{2\pi}{\text{Period}}$

$\text{Period}=\frac{2\pi\times 3}{\pi}$

$\text{Period}=6$

Therefore, The entire cycle take 6 hours.