Growing the Raw Materials As the gate opens, you step into a room that appears to be a laboratory of some kind. Your computer informs you that this is where the raw materials that power the spacecraft are grown. "What do you mean, grown?", you ask. Your computer explains that a special bacteria is grown in this laboratory and converted into fuel. A specic amount of bacteria must be present for the next step in the process to work (within a reasonable margin of error). The computer provides you with information about the population of the bacteria, which follow an exponential model, shown below. In this model, P is the population of bacteria (total count, measured in millions) and t is the time measured in minutes. The population will grow based on this model once a light source is provided and stop immediately once the light source is removed. P(t) = 5et/16 Your computer continues with a note from the les that the alien civilization performed these calculations on the linearization of P(t). Therefore, you will need to linearize P (t) and then use that model to determine when to remove the light source to have 5.145 million bacteria. Your linearization should be around 15 = 0, since this is the time when the light source is switched on. ab % ya |a| 7r sin(a) ill: 0 Using this linearization, after how many minutes (to three decimal places) will you turn off the light to generate a population of 5.145 million bacteria? t = Number minutes