if a stone is thrown into the sea, the distance s(t) it travels after t seconds is s(t) = aln( cosh bt ), where a and b are positive constants. find the velocity v and evaluate v(30)..?..
*how do i differentiate s(t) = aln( cosh bt )..?

Question

if a stone is thrown into the sea, the distance s(t) it travels after t seconds is s(t) = aln( cosh bt ), where a and b are positive constants. find the velocity v and evaluate v(30)..?..
*how do i differentiate s(t) = aln( cosh bt )..?

anonymous · Answer

Use the chain rule.

anonymous · Answer

$$
u(t) = \cosh(bt)\
s(t) = a\ln(u(t)) 
$$

anonymous · Answer

$$
du = \sinh(bt)dt\
ds = \frac{a\;du}{u}
$$

anonymous · Answer

Now, we can substitute for $du$ and $u$: $$
ds = \frac{ab\sinh(bt)\;dt}{\cosh(bt)} = ab\tanh(bt)dt
$$And so $$
s'(t) = \frac{ds}{dt} = ab\tanh(bt)
$$

anonymous · Answer

Should have said $$
du = b\sinh(bt)dt
$$up above.  It was a typo.

anonymous · Answer

then how do i evaluate v(30)..?.. did i simply substitute it..?..

anonymous · Answer

Well, $v(t) = s'(t)$.  So $v(30) = s'(30)$.

anonymous · Answer

so v(30) = abtanh(b(30))..? am i right..?

anonymous · Answer

Yes.

anonymous · Answer

is it possible to evaluate it further.,. i mean making v(30) = a number like 34, 50 or any ather number..?

anonymous · Answer

No matter what you do, the answer would likely still be a function of $a$ and $b$ since they are variables which have not been specified.

anonymous · Answer

I think it is simplified enough as is.  I'm not sure what hyperbolic identities would be used to simplify further.

anonymous · Answer

ok thanks.... by the way how do i send you a medal..?

anonymous · Answer

"Best Answer" I think.

anonymous · Answer

Click on "Best Response"

anonymous · Answer

Blue button in top right corner.

anonymous · Answer

^

anonymous · Answer

do you receive a medal from me... i just click the *best response* button..?

anonymous · Answer

yeah