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The provided image describes a mathematical problem related to the population of bees in a colony. The key details include a differential equation for the rate of change of the population of bees \( p(t) \) with respect to time \( t \), where \( t \) is measured in months. The equation is as follows:

\[
\frac{dp}{dt} = \frac{1}{10} \left( 4 \cdot e^{-0.3t} - 3 \cdot t \cdot e^{-0.3t} \right)
\]

The questions posed in the problem include:

1. Finding the function \( p(t) \) that describes the population of bees in the colony as a function of time.
2. Determining when the population of bees reaches its maximum and minimum.
3. Identifying the intervals of time during which the population increases or decreases.
4. Calculating the population at exactly 2 months and estimating the population at around 2 months.
5. Investigating what happens to the population as time approaches infinity.
6. Finding the variation in population between 2 and 4 months and understanding its significance.

### Steps to approach the problem:

1. **Solve the differential equation** to find the function \( p(t) \).
2. **Analyze the behavior of \( p(t) \)** to identify when it reaches a maximum or minimum.
3. **Calculate the population** at specific points, such as at \( t = 2 \) months.
4. **Examine the population change over time**, including long-term trends and between specific intervals.

I can help solve this step-by-step, starting with finding the function \( p(t) \). Would you like me to proceed with solving the differential equation or a different part first?

An apiculturist analyzed the bee population p(t) (in thousands) in a colony over time t (in months). The rate of change of the population is described by the differential equation:

dp/dt = (1/10) * (4 * e^(-0.3t) - 3 * t * e^(-0.3t))

The questions posed include:
1. Finding the population function p(t) based on the initial population.
2. Determining the time when the population is at its maximum and minimum.
3. Identifying intervals of population growth and decline.
4. Calculating the population at exactly 2 months and estimating near that time.
5. Understanding long-term population trends as time approaches infinity.
6. Determining the population variation between 2 and 4 months and its significance.

Explore the population dynamics of bees using a differential equation model. This problem involves solving a first-order linear differential equation, analyzing maximum and minimum populations, growth intervals, and long-term trends. Suitable for advanced calculus students.

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