Question

The vertical displacement, h(t),in metres, of a cork floating in rough water can be given by h(t)=0.5 cos(2t)+0.7sin(t), where t is the time , in seconds. Determine the maximum velocity that the cork achieves and the first time it occurs

Answer 1

I found the first and second derivative, and then found the zeros of the second derivatives (so, where the velocity is highest) and then do i put those values back into the first derivative equation? Also, one of my zeros for the second derivative was negative so since I'm working with time I should discard that solution, correct?

Answer 2

max/min values occur when first derivative = 0, right ?

Answer 3

set \(h'(t) = 0\)
and solve \(t\)

Answer 4

that gives the critical points

Answer 5

yes. but i'm looking for max and min velocity (first derivative is veolocity) so i'm looking for where the second derivative equals zero.

Answer 6

Ahh I see, yes you're right !

Answer 7

http://www.wolframalpha.com/input/?i=%280.5+cos%282t%29%2B0.7sin%28t%29%29%27%27%3D0%2C+0%3Ct%3C1

Answer 8

okay, I think I have everything right but I have an answers sheet and the answers are a little off so I just wanted to check that I was right.

Answer 9

whats the maximum velocity you're getting ?

Answer 10

my zeros for the second derivative are 0.927 and -0.675 (which i'm not using because negative time isn't a thing), so my max velocity is -0.53m/s

Answer 11

which kind makes sense, because it's not like the cork can ever be out of the water while it's floating, so it can never have a positive value.

Answer 12

im getting max velocity = +1.522, when t = 5.6

Answer 13

my teacher had 5.6 as an asnswer too! can you tell me how you got 5.6?

Answer 14

whats the period of velocity ?

Answer 15

velocity = -sin(2t) + 0.7cos(t)


period = pi + 2pi = 3pi, right ?

Answer 16

so you need to look for ALL critical points in the interval : t = [0, 3pi]

Answer 17

and pick the max value

Answer 18

your -0.53 is just a local minimum value.

Answer 19

To get the Global maximum value :

1) Find all the critical values
2) Take max of above.

Answer 20

since the period of velocity is 3pi,
finding all the critical values in [0, 3pi] is sufficient

Answer 21

http://www.wolframalpha.com/input/?i=maximize+-sin%282t%29+%2B+0.7cos%28t%29%2C+0%3Ct%3C3pi