Question

A culture starts with 230 bacteria. After 30 minutes the number of bacteria is 1150. 5.1 find a formula for the number of bacteria after t hours. 5.2 use the formula to find the number of bacteria after 1 hour. 5.3 after how many minutes will the number of bacteria be 125 times the original number? Leave the answer in terms of ln but in simplified form.

Answer 1

The population $N(t)$ at time $t$ is modeled by the function 

$$N(t)=N(0)e^{kt}$$

5.1. Given $N(0)=230, N(0.5)=1150.$ Substitute

$$1150=230e^{k(0.5)}$$

$$e^{0.5k}=5$$

$$k=2\ln 5$$

$$k=\ln 25$$

$$N(t)=230(25)^{t}$$

5.2.

$$N(1)=230(25)^{1}=5750$$

5.3

$$N(t)=230(25)^{t}=230(125)$$

$$5^{2t}=5^3$$

$$t=1.5$$

$$t=90\ minutes$$