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.
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
Thanks a lot.
Can you tell me where you got:
(t^2)(ln2(t) - 6)
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?
are the answers t=0 or t=6/ln2
To find the initial concentration c(0) at time t=0 min, we can simply substitute t=0 into the equation c(t) = t^3 * 2^(-t/2).
c(0) = 0^3 * 2^(-0/2)
= 0 * 2^0
= 0
Therefore, the initial concentration c(0) at time t=0 min is 0.
To find the maximum concentration of the dye and the time when this occurs, we need to find the critical points of the function c(t) and determine which one corresponds to the maximum value.
To find the critical points, we need to find where the derivative of c(t) is equal to zero.
Let's first find the derivative of c(t):
c'(t) = d/dt (t^3 * 2^(-t/2))
To simplify the differentiation process, we can use the product rule:
c'(t) = (3t^2) * 2^(-t/2) + t^3 * (d/dt(2^(-t/2)))
Now, let's find the derivative of 2^(-t/2) using the chain rule:
d/dt(2^(-t/2)) = (ln(2) * 2^(-t/2)) * (-1/2)
Therefore, substituting the derivative of 2^(-t/2) into the expression for c'(t):
c'(t) = (3t^2) * 2^(-t/2) + t^3 * (ln(2) * 2^(-t/2)) * (-1/2)
Now, to find the critical points, we set c'(t) equal to zero and solve for t:
(3t^2) * 2^(-t/2) + t^3 * (ln(2) * 2^(-t/2)) * (-1/2) = 0
Simplifying the equation:
(3t^2) * 2^(-t/2) - (ln(2) * t^3 * 2^(-t/2)) * (1/2) = 0
Multiplying through by 2^t/2:
(3t^2) - (ln(2) * t^3) * (1/2) = 0
Now, we can solve this equation for t to find the value(s) of t that correspond to the critical point(s) and then substitute the value(s) back into c(t) to determine the maximum concentration.
Unfortunately, it is not possible for me to solve this equation symbolically. However, you can use numerical methods, such as graphing the function or using root-finding algorithms, to approximate the critical point(s) and the maximum concentration of the dye.