## anonymous one year ago MEDAL!!!!! The temperature of a chemical reaction ranges between 20 degrees Celsius and 160 degrees Celsius. The temperature is at its lowest point when t = 0, and the reaction completes 1 cycle during an 8-hour period. What is a cosine function that models this reaction?

1. anonymous

answer choices f(t) = -90 cos pi/4t +70 f(t) = -70 cos pi/4t +90 f(t) = 70 cos 8t +90 f(t)= 90 cos 8t +70

2. anonymous

Let's figure out the amplitude of the cosine function first. Given that the min is 20 degrees and max is 160 degrees, what is the amplitude?

3. anonymous

90

4. anonymous

Amplitude should be calculated as $A = \frac{ Max - Min }{ 2 }$

5. anonymous

70

6. anonymous

Correct, so A and D are gone.

7. anonymous

8. anonymous

So now calculate omega such that $70 \cos(\omega t) + shift$

9. anonymous

We are given the period 8.

10. anonymous

$\omega = \frac{ 2 \pi}{ k }$

11. anonymous

Where k = 8 (period)

12. anonymous

What is omega in this case?

13. anonymous

idk

14. anonymous

$\omega = \frac{ 2\pi }{ 8 } = \frac{ \pi }{ 4 }$

15. anonymous

oh

16. anonymous

17. anonymous

Correct

18. anonymous

19. anonymous

Compare the functions below: f(x) = −3 sin(x − π) + 2 g(x) x y 0 8 1 3 2 0 3 −1 4 0 5 3 6 8 h(x) = (x + 7)^2 − 1

20. anonymous

which function has the smallest minimum

21. anonymous

Let's look at the f(x). We can quickly determine the minimum of the function by looking at its amplitude.

22. anonymous

Without the shift, it would be -3 right? Combining the vertical shift of 2, the minimum is -3+2=-1.

23. anonymous

i believe that h(x) has the smallest minimum value

24. anonymous

The minimum of g(x) is clearly shown in the table as -1.

25. anonymous

h(x) minimum is also -1

26. anonymous

Now, h(x) is a parabola. A parabola is normally centered at the origin, but it has been shifted downwards by -1.

27. anonymous

Correct, all of them have the same minimum.

28. anonymous

thanks