2:00 LTE 324 myopenmath.com Score on last try: 2.5 of 5 pts. See Details for more. Get a similar question You can retry this question below Scientists in a lab are experimenting with some gentically modified amoebas in a petri dish. The amoeba split into triples (so one amoeba becomes 3 separate amoebas) every 15 minutes. There are 150 amoeba in the petri dish at the start of the experiment. NOTE: Any answers in scientific notation should have at least 3 decimal places. What is an equation to model the amount of amoebas in the petri dish in terms of the number of 15 minute periods? $$ A(t)=150 \cdot 3^{t} \quad \vee \sigma^{\infty} $$ How many amoeba are in the dish after 1 hour? $$ \square $$ 12150 $$ \square $$ How many amoeba are in the dish after 5 hours? $$ \square $$ How many amoeba are in the dish after 1 day? (How many hours is this?) $$ \square $$

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UpStudy AI · Accepted Answer

Let's work through each part of the problem step-by-step.

### **Problem Statement:**
- **Initial Population:** 150 amoebas.
- **Growth Rate:** Every 15 minutes, each amoeba splits into 3, meaning the population triples every 15 minutes.
- **Objective:** 
  1. Formulate an equation to model the amoeba population over time.
  2. Calculate the population after specific time intervals.

### **1. Formulating the Equation**

Let:
- $$ t $$ = Number of 15-minute periods.
- $$ A(t) $$ = Number of amoebas after $$ t $$ periods.

Since the population triples every 15 minutes:
$$
A(t) = 150 \times 3^t
$$

**Equation:**  
$$
A(t) = 150 \cdot 3^{t}
$$

### **2. Population After Specific Time Intervals**

#### **a. After 1 Hour**

- **Convert Hours to 15-Minute Periods:**
  $$
  1 \text{ hour} = 60 \text{ minutes} \Rightarrow t = \frac{60}{15} = 4 \text{ periods}
  $$
  
- **Calculate $$ A(4) $$:**
  $$
  A(4) = 150 \times 3^4 = 150 \times 81 = 12,\!150
  $$
  
**Answer:**  
$$
\boxed{12,\!150}
$$

#### **b. After 5 Hours**

- **Convert Hours to 15-Minute Periods:**
  $$
  5 \text{ hours} = 300 \text{ minutes} \Rightarrow t = \frac{300}{15} = 20 \text{ periods}
  $$
  
- **Calculate $$ A(20) $$:**
  $$
  A(20) = 150 \times 3^{20}
  $$
  
- **Compute $$ 3^{20} $$:**
  $$
  3^{20} = 3,486,784,401
  $$
  
- **Multiply by 150:**
  $$
  A(20) = 150 \times 3,486,784,401 = 523,\!017,\!660,\!150
  $$
  
- **Express in Scientific Notation:**
  $$
  523,\!017,\!660,\!150 = 5.231 \times 10^{11}
  $$
  
**Answer:**  
$$
\boxed{5.231 \times 10^{11}}
$$

#### **c. After 1 Day**

- **Convert Days to Hours:**
  $$
  1 \text{ day} = 24 \text{ hours}
  $$
  
- **Convert Hours to 15-Minute Periods:**
  $$
  24 \text{ hours} \times \frac{4 \text{ periods}}{1 \text{ hour}} = 96 \text{ periods}
  $$
  
- **Calculate $$ A(96) $$:**
  $$
  A(96) = 150 \times 3^{96}
  $$
  
- **Compute $$ 3^{96} $$:**
  $$
  3^{96} \approx 5.48 \times 10^{45}
  $$
  
- **Multiply by 150:**
  $$
  A(96) = 150 \times 5.48 \times 10^{45} = 8.22 \times 10^{47}
  $$
  
**Answer:**  
$$
\boxed{8.220 \times 10^{47}}
$$

### **Summary of Answers:**
1. **Equation:**  
   $$
   A(t) = 150 \cdot 3^{t}
   $$
2. **After 1 Hour:**  
   $$
   12,\!150
   $$
3. **After 5 Hours:**  
   $$
   5.231 \times 10^{11}
   $$
4. **After 1 Day:**  
   $$
   8.220 \times 10^{47}
   $$
**Equation to Model the Amoeba Population:** $$ A(t) = 150 \cdot 3^{t} $$ **Population After Specific Time Intervals:** - **After 1 Hour:** 12,150 amoebas - **After 5 Hours:** $$5.231 \times 10^{11}$$ amoebas - **After 1 Day:** $$8.220 \times 10^{47}$$ amoebas

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