The population of a bacterial colony t hours after observation begins is found to be changing at the rate dP/dt = 200e^0.1t+150e^-0.03t. If the population was 200,000 bacteria when observations began, what will the population be 12 hours later?
what i did:
(1/12-0)[antiderivative 200e^0.1t+150e^-0.03t]
(1/12)[(200/1.1)e^1.1t + (150/0.97)e^0.97t]
(1/12)[((200/1.1)e^13.2 + (150/0.97)e^11.64) - ((200/1.1)e^0 + (150/0.97)e^0)]
9650594.820
is this right?? seems like a big number!
Originally Posted by yess
I don't know where you get the 1 and 12 from, if you assume the time observations began is time 0 your limits are 12 and 0.
You've also got the integration wrong, when talking about exponentials we do not add 1 to the power:which can be shown by taking the derivative using the chain rule
![\displaystyle \int^{12}_0 200e^{0.1t}+150e^{-0.03t}~dt = \left[2000e^{0.1t}-\frac{100}{3}\times 150e^{-0.03t}\right]^{12}_0](http://latex.codecogs.com/png.latex?\displaystyle \int^{12}_0 200e^{0.1t}+150e^{-0.03t}~dt = \left[2000e^{0.1t}-\frac{100}{3}\times 150e^{-0.03t}\right]^{12}_0)
so using your equation I sub in 12 for t and then I sub in 0 for t and subtract that from the value where t = 12? This gives me 6151.852215 so does this mean the population 12 hours later is about 6151.85 bacteria?
I don't know where you get the 1 and 12 from, if you assume the time observations began is time 0 your limits are 12 and 0.
You've also got the integration wrong, when talking about exponentials we do not add 1 to the power:
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