anonymous 4 years ago A spring hanging from the ceiling vibrates up and down. it's vertical position at time (t) is given by f(t)=4 sin(3t). (a) Find the velocity of the spring at time t. (b) What is the spring's maximum speed. (c) what is location when it reaches its maximum speed.

1. anonymous

You know that the position as a function of time is: $x(t) = 4\sin(3t)$ you also know that velocity is defined as: $\frac{dx}{dt}$

2. anonymous

can you find the derivative of: $x(t) = 4\sin(3t)$

3. anonymous

Yes that is the ans of a.what about b and c parts?

4. anonymous

so we want to find the maximum of: $12\cos(3t)$

5. anonymous

So this basically comes down to finding where the cosine function has it's maximums

6. anonymous

and then using the value for the maximum of a cosine in the equation

7. anonymous

So, what is the maximum of a cosine function?

8. anonymous

1

9. anonymous

yes, so the maximum value for the speed must be:

10. anonymous

i think 12

11. anonymous

yes

12. anonymous

At what time does the speed become 12?

13. anonymous

don't know.may be 0

14. anonymous

Yes, because cos(x) = 1 when x = 0

15. anonymous

Now that you know the time at which it reaches its maximum speed, you can plug that into the first equation

16. anonymous

which first equation?

17. anonymous

the equation for position as a function of time

18. anonymous

f(t)=4sin(3t)

19. anonymous

yes

20. anonymous

what i do with this now? please explain

21. anonymous

i put t=0 so position is zero?

22. anonymous

Yes

23. anonymous

ok.Thanks a lot.

24. anonymous

You are welcome