rate of change in number of bacteria is proportional to....

[math]

the rate of change in the number of bacteria present in a culture is proportional to twice the number of bacteria present. If y(0) = 6 and y(2) = 14, find y(5).

I'm not sure if I did this problem correctly:

dy/dt = 2ky
dy = k * 2y * dt
integral of dy/2y = integral of k dt
(1/2) ln abs(y) = kt + c
ln abs(y) = 2kt + 2c
y(t) = e ^ (2kt + 2c) = e^(2kt) * e^(2c)
y(t) = ce^(2kt)
y(0) = 6 = ce^0 c= 6
y(2) = 14 = 6e^(4k)
7/3 = e^(4k)
ln (7/3) = 4k
k = .2118
y(t) = 6e^.424t)
y(5) = 6e^(.424 * 5)
= 49.899

 

[galactus]

Looks good and good work.

Maybe leave it as \(\displaystyle \L\\y=6e^{\left(\frac{ln(\frac{7}{3})}{2}t\right)}\).

Looks better than decimals, I think.

 

[morson]

y = 6 * (7/3)^(t/2) looks even better!