| Day | Number of bacteria per milliliter at end of day |
| 1 |
|
| 2 |
|
| 3 |
|
A culture of bacteria is growing at an exponential rate, as shown in the table above. At this rate, on which day would the number of bacteria per milliliter reach ?
Correct answer: D
Explanation
Choice D is correct. The number of bacteria per milliliter is doubling each day. For example, from day 1 to day 2, the number of bacteria increased from 2.5 × 105 to 5.0 × 105. At the end of day 3 there are 106 bacteria per milliliter. At the end of day 4, there will be 106 × 2 bacteria per milliliter. At the end of day 5, there will be (106 × 2) × 2, or 106 × (22) bacteria per milliliter, and so on. At the end of day d, the number of bacteria will be 106 × (2d – 3). If the number of bacteria per milliliter will reach 5.12 × 108 at the end of day d, then the equation must hold. Since 5.12 × 108 can be rewritten as 512 × 106, the equation is equivalent to
. Rewriting 512 as 29 gives d – 3 = 9, so d = 12. The number of bacteria per milliliter would reach 5.12 × 108 at the end of day 12.
Choices A, B, and C are incorrect. Given the growth rate of the bacteria, the number of bacteria will not reach 5.12 × 108 per milliliter by the end of any of these days.
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