8. En una tarea de ciencias se te pide hacer un experimento, a fin de que detectes el comportamiento en el crecimiento de población de ácaros. Tomas una muestra de una caja de vidrio de laboratorio (petri), la cual contiene una población inicial de mil ácaros, proporcionas alimento y registras a diario el crecimiento de la población. El registro de los tres días, fue el siguiente:
[tex]
\begin{array}{l}
1 \text { día }=4000 \\
2 \text { días }=7000 \\
3 \text { días }=10000
\end{array}
[/tex]
De acuerdo con este comportamiento, ¿cuál es el modelo matemático que lo representa?
A) [tex]$y=3000 x+1000$[/tex]
B) [tex]$y=1000 x+3000$[/tex]
C) [tex]$y=x^3+1000 x$[/tex]
D) [tex]$y=3000^x$[/tex]
Answer (1)
The problem provides the population of mites over three days and asks for the mathematical model representing the growth.
Calculate the difference between consecutive population values to check for linear growth.
Confirm that the growth is linear with a constant difference of 3000.
Determine the equation of the line as y = 3000 x + 1000 .
The correct model is y = 3000 x + 1000 .
Explanation
Analyzing the Problem We are given the initial population of mites and the population after 1, 2, and 3 days. We need to find the mathematical model that represents this growth. Let's analyze the data to see if the growth is linear or exponential.
Checking for Linear Growth Let x be the number of days and y be the population of mites. We have the following data:
Initial population ( x = 0 ): y = 1000
After 1 day ( x = 1 ): y = 4000
After 2 days ( x = 2 ): y = 7000
After 3 days ( x = 3 ): y = 10000
Let's check if the growth is linear by calculating the difference between consecutive population values.
Confirming Linear Growth The difference between the population after 1 day and the initial population is:
4000 − 1000 = 3000
The difference between the population after 2 days and the population after 1 day is:
7000 − 4000 = 3000
The difference between the population after 3 days and the population after 2 days is:
10000 − 7000 = 3000
Since the difference is constant, the growth is linear with a slope of 3000.
Determining the Equation The equation of the line is of the form y = m x + b , where m is the slope and b is the y-intercept. In this case, m = 3000 and b is the initial population, which is 1000. Therefore, the equation is:
y = 3000 x + 1000
Selecting the Correct Option Now, let's compare this model with the given options:
A) y = 3000 x + 1000 B) y = 1000 x + 3000 C) y = x 3 + 1000 x D) y = 300 0 x
The correct model is option A.
Final Answer Therefore, the mathematical model that represents the growth of the mite population is y = 3000 x + 1000 .
Examples
Linear growth models are useful in many real-world scenarios. For example, if you deposit a fixed amount of money into a savings account each month, the total amount of money in the account will grow linearly over time. Similarly, if you are driving at a constant speed, the distance you travel will increase linearly with time. Understanding linear growth can help you make predictions and plan for the future.
