Question

c(t) = t^3 * 2^(-t/2)

Find the initial concentration c(0) at time t=0 min.
Finds the maximum concentration of the dye sand the time when this occurs.

Answer 1

first part is trivial, just put t = 0 into the function

c'(t) = t^3(-1/2)ln2(2^(-t/2) + 3t^2(2^(-t/2)
= 2^(-t/2)((-1/2)ln2(t^3) + 3t^2)
= 0 for max
2^(-t/2) = 0 ---> no solution
or
(t^2)(ln2(t) - 6)

t = 0, or t = 6/ln2

clearly t = 0 gives zero concentration, so
t = 6/ln2 = 8.656
C(8.656) = 32.29

Answer 2

Thanks a lot.
Can you tell me where you got:
(t^2)(ln2(t) - 6)

Answer 3

from
(-1/2)ln2(t^3) + 3t^2
we are setting that equal to 0

(-1/2)ln2(t^3) + 3t^2 = 0
multiply by -2
ln2(t^3) - 6t^2 = 0
t^2( ln2(t) - 6) = 0 etc.

ok?

Answer 4

are the answers t=0 or t=6/ln2