QQuestionAnatomy and Physiology
QuestionAnatomy and Physiology
Growth of a Culture of Bacteria
| Day | Number of bacteria per
milliliter at end of day |
| --- | --- |
| 1 | $2.5 \times 10^{5}$ |
| 2 | $5.0 \times 10^{5}$ |
| 3 | $1.0 \times 10^{6}$ |
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 $5.12 \times 10^{6}$ ? A. Day 5 B. Day 9 C. Day 11 D. Day 12
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Answer
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Step 1:: Identify the formula for exponential growth
where $N(t)$ is the number of bacteria at time t, $N_0$ is the initial number of bacteria, $growth\ rate$ is the multiplication factor per unit time, and t is the time in days.
The formula for exponential growth is given by:
Step 2:: Determine the initial number of bacteria, $N_1$
N_0 = 2.5 \times 10^{5}
From the table, the initial number of bacteria on Day 1 is:
Step 3:: Determine the growth rate
growth\ rate = 2
We can find the growth rate by comparing the number of bacteria between any two consecutive days. For example, between Day 1 and Day 2, the number of bacteria increases by a factor of: Since the growth is exponential, this factor remains constant over time. Therefore, the growth rate is:
Step 4:: Find the time, t, when the number of bacteria reaches $5.12 \times 10^{6}$
t = 5
We need to solve the equation: Next, simplify the right-hand side: Now, take the base- 2 logarithm of both sides to find t: Using a calculator, we find: Since time is measured in days, we round up to the nearest whole day:
Step 5:: Identify the day that satisfies the condition
The number of bacteria reaches $5.12 \times 10^{6}$ on Day 5.
Final Answer
The number of bacteria reaches $5.12 \times 10^{6}$ per milliliter on Day 5.
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