Answer:
The maximum amount of red kryptonite present is 33.27 g after 4.93 hours.
Step-by-step explanation:
dy/dt = y(1/t - k)
separating the variables, we have
dy/y = (1/t - k)dt
dy/y = dt/t - kdt
integrating both sides, we have
∫dy/y = ∫dt/t - ∫kdt
㏑y = ㏑t - kt + C
㏑y - ㏑t = -kt + C
㏑(y/t) = -kt + C
taking exponents of both sides, we have
[tex]\frac{y}{t} = e^{-kt + C} \\\frac{y}{t} = e^{-kt}e^{C} \\\frac{y}{t} = Ae^{-kt} (A = e^{C})\\y = Ate^{-kt}[/tex]
when t = 1 hour, y = 15 grams. So,
[tex]y = Ate^{-kt}\\15 = A(1)e^{-kX1}\\15 = Ae^{-k}[/tex](1)
when t = 3 hours, y = 30 grams. So,
[tex]y = Ate^{-kt}\\30 = A(3)e^{-kX3}\\30 = 3Ae^{-3k}[/tex] (2)
dividing (2) by (1), we have
[tex]\frac{30}{15} = \frac{3Ae^{-3k}}{Ae^{-k}} \\2 = 3e^{-2k}\\\frac{2}{3} = e^{-2k}[/tex]
taking natural logarithm of both sides, we have
-2k = ㏑(2/3)
-2k = -0.4055
k = -0.4055/-2
k = 0.203
From (1)
[tex]A = 15e^{k} \\A = 15e^{0.203} \\A = 15 X 1.225\\A = 18.36[/tex]
Substituting A and k into y, we have
[tex]y = 18.36te^{-0.203t}[/tex]
The maximum value of y is obtained when dy/dt = 0
dy/dt = y(1/t - k) = 0
y(1/t - k) = 0
Since y ≠ 0, (1/t - k) = 0.
So, 1/t = k
t = 1/k
So, the maximum value of y is obtained when t = 1/k = 1/0.203 = 4.93 hours
[tex]y = 18.36(1/0.203)e^{-0.203t}\\y = \frac{18.36}{0.203}e^{-0.203X1/0.203}\\y = 90.44e^{-1}\\y = 90.44 X 0.3679\\y = 33.27 g[/tex]
So the maximum amount of red kryptonite present is 33.27 g after 4.93 hours.
