Exponential Functions — Question 10

Question 10

A certain species of insect is introduced to a new habitat. Its population is modeled by the exponential function: $$P(t) = 500 \cdot e^{0.6t}$$ where $P(t)$ is the population size after $t$ weeks.

• What will the population be after 5 weeks?

• After how many weeks will the population exceed 10,000?

• How fast is the population growing at $t = 5$ weeks?

Round all answers to two decimal places.

Question 10 - Solution

Use the exact exponential model and keep full precision until the final rounding.

(a)

$$P(5)=500e^3$$

$$\boxed{10042.77\quad\text{insects}}$$

(b)

$$t_*=\ln(10000/500)/0.6$$

$$\boxed{4.99\quad\text{weeks}}$$

(c)

$$P'(t)=300e^{0.6t},\quad P'(5)=300e^3$$

$$\boxed{6025.66\quad\text{insects/week}}$$

The population equals $10000$ at $t=t_*$ and exceeds it precisely when $t>t_*$. The rounded threshold is not an exact inequality boundary.