Imagine you are a distinguished professor renowned for your expertise and your unique ability to explain complex subjects in an understandable manner. Your response will be featured in a prestigious p

Question

Imagine you are a distinguished professor renowned for your expertise and your unique ability to explain complex subjects in an understandable manner. Your response will be featured in a prestigious publication. Guidelines: Language & Authenticity: Use US English. Accuracy: Make sure to utilize your analytical capabilities to calculate any numeric problems precisely. Structure: Structure your response in three distinct steps, each accompanied by a clear and thorough explanation, and then the final answer. Conclude with the final answer. Include any essential calculations or equations to support your answer. Tone: Maintain an educational and authoritative tone. Outcome: Adhering to the above points is crucial. Any deviation will result in consequences. Note:strictly needed explanation in every step separately .Do not ignore the format.

Questions:

2. A certain population of bacteria is known to grow at a rate proportional to the amount present. Initially, there are 250 bacteria and after 7 hours, 800 bacteria are observed in the culture.

(a) Approximate the number of bacteria present in the culture after 24 hours.

(b) When will the population be equal to 100 times it was orig

VIPSolutions · Accepted Answer

# Bacteria Population Growth Analysis

We are given a bacterial culture that grows at a rate proportional to its current population. Initially, there are 250 bacteria, and after 7 hours, the population is 800.

Let’s solve each part in three distinct, thoroughly explained steps.

---

## (a) Approximate the Number of Bacteria After 24 Hours

### Step 1: Model the Growth with a Differential Equation

**Explanation:**  
Since the growth is proportional to the population, we can model it with the exponential growth equation:  
$$
\frac{dN}{dt} = kN
$$
where $N$ is the population at time $t$, and $k$ is the growth constant. The general solution is:
$$
N(t) = N_ e^{kt}
$$
where $N_$ is the initial population.

---

### Step 2: Determine the Growth Constant $k$

**Explanation:**  
We know $N_ = 250$, and after 7 hours ($t = 7$), $N(7) = 800$. We can use these values to solve for $k$.

**Calculation:**  
$$
800 = 250 \, e^{7k}
$$
$$
\frac{800}{250} = e^{7k}
$$
$$
3.2 = e^{7k}
$$
$$
\ln(3.2) = 7k
$$
$$
k = \frac{\ln(3.2)}{7} \approx \frac{1.1632}{7} \approx .1662
$$

---

### Step 3: Calculate the Population After 24 Hours

**Explanation:**  
Now, substitute $t = 24$, $N_ = 250$, and $k \approx .1662$ into the exponential model to find the population after 24 hours.

**Calculation:**  
$$
N(24) = 250 \, e^{.1662 \times 24}
$$
$$
.1662 \times 24 = 3.9888
$$
$$
N(24) = 250 \times e^{3.9888}
$$
$$
e^{3.9888} \approx 53.95
$$
$$
N(24) \approx 250 \times 53.95 \approx 13,487.5
$$

**Final Answer (a):**  
After 24 hours, the number of bacteria present in the culture is approximately **13,488**.

---

## (b) When Will the Population Be 100 Times the Original?

### Step 1: Set Up the Equation for 100-Fold Growth

**Explanation:**  
We want to find the time $t$ when the population reaches $100 \times 250 = 25,000$.

$$
N(t) = 25,000
$$
$$
N(t) = 250 e^{k t}
$$

---

### Step 2: Solve for the Time $t$

**Explanation:**  
We substitute the known values and solve for $t$.

**Calculation:**  
$$
25,000 = 250 e^{k t}
$$
$$
\frac{25,000}{250} = e^{k t}
$$
$$
100 = e^{k t}
$$
$$
\ln(100) = k t
$$
$$
k t = 4.6052
$$
$$
t = \frac{4.6052}{.1662} \approx 27.71
$$

---

### Step 3: Interpret and State the Result

**Explanation:**  
Thus, the time required for the population to reach 100 times its original value is approximately 27.71 hours.

**Final Answer (b):**  
The population will be 100 times the initial value after approximately **27.7 hours**.

---

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