The size $$ P $$ of a certain insect population at time $$ t $$ (in days) obeys the function $$ P(t)=600 e^{0.04 t} $$ (a) Determine the number of insects at $$ t=0 $$ days, (b) What is the growth rate of the insect population? (c) Graph the function using a graphing utility. (d) What is the population after 10 days? (e) When will the insect population reach 720 ? (f) When will the insect population double?

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Let's analyze the given population model step by step.

The population $$ P(t) $$ at time $$ t $$ (in days) is given by:
$$ P(t) = 600 e^{0.04 t} $$

### **a) Determine the number of insects at $$ t = 0 $$ days**

To find the initial population at $$ t = 0 $$, substitute $$ t = 0 $$ into the function:
$$
P(0) = 600 e^{0.04 \times 0} = 600 e^{0} = 600 \times 1 = 600
$$
**Answer:** The initial number of insects at $$ t = 0 $$ days is **600**.

---

### **b) What is the growth rate of the insect population?**

The general form of an exponential growth function is:
$$
P(t) = P_0 e^{rt}
$$
where:
- $$ P_0 $$ is the initial population,
- $$ r $$ is the growth rate,
- $$ t $$ is time.

Comparing this with the given function:
$$
P(t) = 600 e^{0.04 t}
$$
We can see that the growth rate $$ r $$ is **0.04 per day**.

**Answer:** The growth rate of the insect population is **0.04** (or **4% per day**).

---

### **c) Graph the function using a graphing utility**

While I can't produce a graph directly here, I can guide you on how to graph the function $$ P(t) = 600 e^{0.04 t} $$ using common tools:

**Using Desmos (Online Graphing Calculator):**
1. Go to [Desmos Graphing Calculator](https://www.desmos.com/calculator).
2. In the input field, type: `P(t) = 600 * e^(0.04 * t)`
3. Press Enter to see the graph.

**Using Microsoft Excel:**
1. Create a table with values of $$ t $$ (e.g., from 0 to 100 days).
2. Calculate $$ P(t) $$ for each $$ t $$ using the formula `=600*EXP(0.04*A2)` assuming $$ t $$ is in cell A2.
3. Select the data and insert a scatter plot with a smooth curve.

**Graph Characteristics:**
- **Initial Value:** Starts at 600 when $$ t = 0 $$.
- **Growth:** Exponential increase; the curve becomes steeper over time.
- **Asymptote:** $$ P(t) $$ increases without bound as $$ t $$ increases.

---

### **d) What is the population after 10 days?**

To find the population at $$ t = 10 $$ days, substitute $$ t = 10 $$ into the function:
$$
P(10) = 600 e^{0.04 \times 10} = 600 e^{0.4}
$$
Calculate $$ e^{0.4} $$:
$$
e^{0.4} \approx 1.49182
$$
Now, multiply by 600:
$$
P(10) \approx 600 \times 1.49182 \approx 895.09
$$
**Answer:** After 10 days, the population is approximately **895 insects**.

---

### **e) When will the insect population reach 720?**

We need to solve for $$ t $$ when $$ P(t) = 720 $$:
$$
720 = 600 e^{0.04 t}
$$
First, divide both sides by 600:
$$
\frac{720}{600} = e^{0.04 t} \quad \Rightarrow \quad 1.2 = e^{0.04 t}
$$
Take the natural logarithm of both sides:
$$
\ln(1.2) = 0.04 t
$$
Solve for $$ t $$:
$$
t = \frac{\ln(1.2)}{0.04}
$$
Calculate $$ \ln(1.2) $$:
$$
\ln(1.2) \approx 0.1823216
$$
Now, compute $$ t $$:
$$
t \approx \frac{0.1823216}{0.04} \approx 4.55804 \text{ days}
$$
**Answer:** The population will reach 720 insects after approximately **4.56 days**.

---

### **f) When will the insect population double?**

To find when the population doubles, set $$ P(t) = 2P_0 = 2 \times 600 = 1200 $$:
$$
1200 = 600 e^{0.04 t}
$$
Divide both sides by 600:
$$
2 = e^{0.04 t}
$$
Take the natural logarithm of both sides:
$$
\ln(2) = 0.04 t
$$
Solve for $$ t $$:
$$
t = \frac{\ln(2)}{0.04}
$$
Calculate $$ \ln(2) $$:
$$
\ln(2) \approx 0.6931472
$$
Now, compute $$ t $$:
$$
t \approx \frac{0.6931472}{0.04} \approx 17.32868 \text{ days}
$$
**Answer:** The insect population will double after approximately **17.33 days**.

---

**Summary of Answers:**
- **a)** 600 insects
- **b)** 4% per day
- **c)** [Graph described above]
- **d)** Approximately 895 insects after 10 days
- **e)** Approximately 4.56 days to reach 720 insects
- **f)** Approximately 17.33 days to double the population
- **a)** 600 insects at $$ t = 0 $$ days - **b)** Growth rate is 4% per day - **c)** Graph shows exponential growth starting at 600 insects - **d)** Population after 10 days is approximately 895 insects - **e)** Population reaches 720 insects after about 4.56 days - **f)** Population doubles after approximately 17.33 days

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