The size P of a certain insect population at time t (in days) obeys the function \( \mathrm{P}(\mathrm{t})=600 e^{0.04 t} \) (a) Determine the number of insects at \( \mathrm{t}=0 \) 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?

Let's solve each part of the problem step by step. Given the population function: \[ \mathrm{P}(\mathrm{t}) = 600 e^{0.04 t} \] where: - \( \mathrm{P}(\mathrm{t}) \) is the population at time \( \mathrm{t} \) days. - \( e \) is the base of the natural logarithm. - \( 0.04 \) is the growth rate per day. --- ### **a) Determine the number of insects at \( \mathrm{t} = 0 \) days.** To find the initial population (\( t = 0 \)): \[ \mathrm{P}(0) = 600 e^{0.04 \times 0} = 600 e^{0} = 600 \times 1 = 600 \] **Answer:** There are **600 insects** at \( t = 0 \) days. --- ### **b) What is the growth rate of the insect population?** The growth rate is the coefficient of \( t \) in the exponent of the exponential function. \[ \text{Growth Rate} = 0.04 \ \text{per day} \] To express it as a percentage: \[ 0.04 \times 100\% = 4\% \ \text{per day} \] **Answer:** The population grows at a rate of **4% per day**. --- ### **c) Graph the function using a graphing utility.** While I can't provide a visual graph here, I can describe how to graph the function: 1. **Axes:** - **Horizontal Axis (x-axis):** Time \( t \) in days. - **Vertical Axis (y-axis):** Population \( \mathrm{P}(t) \). 2. **Plotting Points:** - Start at \( t = 0 \), \( \mathrm{P}(0) = 600 \). - As \( t \) increases, the population grows exponentially. 3. **Behavior:** - The graph will show an upward-sloping curve that becomes steeper over time, reflecting exponential growth. 4. **Example Points:** - \( t = 0 \), \( \mathrm{P} = 600 \) - \( t = 10 \), \( \mathrm{P} \approx 895 \) (from part d) - \( t \) when \( \mathrm{P} = 1200 \) (double population, from part f) **Answer:** *[A descriptive explanation of the graph is provided. Use a graphing calculator or software to plot \( \mathrm{P}(t) = 600 e^{0.04 t} \) with \( t \) on the x-axis and \( \mathrm{P}(t) \) on the y-axis, showing exponential growth starting at 600.]* --- ### **d) What is the population after 10 days?** Calculate \( \mathrm{P}(10) \): \[ \mathrm{P}(10) = 600 e^{0.04 \times 10} = 600 e^{0.4} \] First, compute \( e^{0.4} \): \[ e^{0.4} \approx 1.49182 \] Then, \[ \mathrm{P}(10) \approx 600 \times 1.49182 \approx 895.09 \] **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 \( \mathrm{P}(t) = 720 \): \[ 720 = 600 e^{0.04 t} \] 1. **Divide both sides by 600:** \[ \frac{720}{600} = e^{0.04 t} \] \[ 1.2 = e^{0.04 t} \] 2. **Take the natural logarithm of both sides:** \[ \ln(1.2) = 0.04 t \] 3. **Solve for \( t \):** \[ t = \frac{\ln(1.2)}{0.04} \] Calculate \( \ln(1.2) \): \[ \ln(1.2) \approx 0.1823216 \] Then, \[ t \approx \frac{0.1823216}{0.04} \approx 4.55804 \ \text{days} \] **Answer:** The population reaches **720 insects** after approximately **4.56 days**. --- ### **f) When will the insect population double?** The initial population is \( \mathrm{P}(0) = 600 \), so double the population is \( 1200 \). Set \( \mathrm{P}(t) = 1200 \): \[ 1200 = 600 e^{0.04 t} \] 1. **Divide both sides by 600:** \[ 2 = e^{0.04 t} \] 2. **Take the natural logarithm of both sides:** \[ \ln(2) = 0.04 t \] 3. **Solve for \( t \):** \[ t = \frac{\ln(2)}{0.04} \] Calculate \( \ln(2) \): \[ \ln(2) \approx 0.693147 \] Then, \[ t \approx \frac{0.693147}{0.04} \approx 17.3287 \ \text{days} \] **Answer:** The population **doubles** after approximately **17.33 days**. --- **Summary of Answers:** - **a)** 600 insects at \( t = 0 \) days. - **b)** Growth rate of 4% per day. - **c)** *Described the graph; use graphing tools to visualize exponential growth.* - **d)** Approximately 895 insects after 10 days. - **e)** Population reaches 720 insects after about 4.56 days. - **f)** Population doubles after approximately 17.33 days.