e). A bacterial culture starts with 1000 bacteria and triples after 5 hours. Find the doubling
period and the amount of bacteria after 8 hours.

Question

e). A bacterial culture starts with 1000 bacteria and triples after 5 hours. Find the doubling period and the amount of bacteria after 8 hours.

UpStudy AI · Accepted Answer

To solve the problem, we'll model the bacterial growth using exponential functions.

### Given:
- **Initial number of bacteria (N₀):** 1,000
- **Triples every 5 hours:** This means after 5 hours, the number of bacteria becomes $$ 3 \times 1,000 = 3,000 $$.

### 1. **Finding the Doubling Period**

The doubling period ($$ t_d $$) is the time it takes for the number of bacteria to double.

**Step 1:** Establish the exponential growth model:
$$ N(t) = N₀ \times e^{kt} $$
where:
- $$ N(t) $$ is the number of bacteria at time $$ t $$,
- $$ k $$ is the growth rate constant.

**Step 2:** Using the information that the bacteria triples in 5 hours:
$$ 3N₀ = N₀ \times e^{5k} $$
$$ 3 = e^{5k} $$
$$ \ln(3) = 5k $$
$$ k = \frac{\ln(3)}{5} $$

**Step 3:** The doubling time ($$ t_d $$) is given by:
$$ t_d = \frac{\ln(2)}{k} $$
$$ t_d = \frac{\ln(2)}{\ln(3)/5} $$
$$ t_d = 5 \times \frac{\ln(2)}{\ln(3)} $$

**Calculating Numerically:**
$$ \ln(2) \approx 0.6931 $$
$$ \ln(3) \approx 1.0986 $$
$$ t_d \approx 5 \times \frac{0.6931}{1.0986} \approx 3.15 \text{ hours} $$

### 2. **Calculating the Number of Bacteria After 8 Hours**

**Using the exponential growth model:**
$$ N(t) = 1,000 \times 3^{t/5} $$

**For $$ t = 8 $$ hours:**
$$ N(8) = 1,000 \times 3^{8/5} $$
$$ N(8) \approx 1,000 \times 5.805 $$
$$ N(8) \approx 5,805 \text{ bacteria} $$

### **Final Answer:**
- **Doubling Period:** Approximately **3.15 hours**
- **Number of Bacteria After 8 Hours:** Approximately **5,805 bacteria**
The doubling period is approximately 3.15 hours, and after 8 hours, there are about 5,805 bacteria.

e). A bacterial culture starts with 1000 bacteria and triples after 5 hours. Find the doubling
period and the amount of bacteria after 8 hours.

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