# The Growth Rate of Bacteria Population in a Petri Dish after 4 Hours

Understanding the growth rate of bacteria is crucial in many fields, including microbiology, medicine, and environmental science. The growth rate of bacteria can be represented mathematically, allowing scientists to predict population sizes at different points in time. In this article, we will explore the growth rate of a bacteria population in a Petri dish after 4 hours, using the function B(t) = 2t^3 + 5t^2 + t + 2. This function represents the number of bacteria in a Petri dish after t hours.

## Understanding the Bacterial Growth Function

The function B(t) = 2t^3 + 5t^2 + t + 2 is a cubic function, which means it describes a curve on a graph. The variable t represents time in hours, and B(t) represents the number of bacteria at time t. The coefficients 2, 5, and 1 in front of the t terms represent the rates at which the bacteria population grows.

## Calculating the Growth Rate after 4 Hours

To find out how fast the population is growing after 4 hours, we need to calculate the derivative of the function B(t). The derivative of a function gives us the rate of change of the function at any given point. In this case, it will give us the rate of growth of the bacteria population at any given time.

The derivative of B(t) = 2t^3 + 5t^2 + t + 2 is B'(t) = 6t^2 + 10t + 1. This new function B'(t) represents the growth rate of the bacteria population at time t.

Substituting t = 4 into B'(t), we get B'(4) = 6(4)^2 + 10(4) + 1 = 112. This means that after 4 hours, the bacteria population in the Petri dish is growing at a rate of 112 bacteria per hour.

## Implications of the Growth Rate

The growth rate of bacteria is an important factor in many scientific and medical contexts. For example, understanding how fast bacteria grow can help in the development of antibiotics and other treatments. It can also help in predicting and controlling the spread of bacterial infections.

In environmental science, knowing the growth rate of bacteria can help in understanding and managing biological processes such as decomposition and nutrient cycling.

In conclusion, the function B(t) = 2t^3 + 5t^2 + t + 2 and its derivative B'(t) = 6t^2 + 10t + 1 provide valuable insights into the growth rate of a bacteria population in a Petri dish. After 4 hours, the population is growing at a rate of 112 bacteria per hour.