Question

A population of beetles is growing at a rate given by the function

dP/dt = 1425e⁻⁰.²⁵/(1 + 3e⁻⁰.²⁵t)².

Where dP/dt is measured in beetles per week and t in weeks.

Starting at time t=0, estimate how long it takes for the beetle population increase by 150 beetles. Be accurate to one decimal digit.

Answer 1

The solution requires numerically integrating the beetle population's growth rate function from t=0 until the integral equals 150, which signifies the population increase. This will yield the time taken for the increase of 150 beetles.

Step-by-step explanation:

The question asks how long it will take for a population of beetles, growing according to a certain function, to increase by 150 beetles. To solve this, we must integrate the given growth rate function with respect to time and find when the population has increased by 150 beetles from the initial population.

To estimate the time t when the population increase of beetles is exactly 150, we set up the integral of the growth rate function from t=0 to t=T, where T is the time we want to find:

\( \int_0^T \frac{{dP}}{{dt}} dt = 150 \)

Plugging in the growth rate function, we get:

\( \int_0^T 1425e^{-0.25t}/(1 + 3e^{-0.25t})^2 dt = 150 \)

We'll have to numerically integrate this equation since it doesn't have a straightforward antiderivative. We'll use numerical methods (like the trapezoidal rule, Simpson's rule, or numerical integration software) to find the value of T to one decimal place.

Once T is found, we would have the time estimate for when the population of beetles increases by 150.