Answer :
The patterns in the data and the analysis, an exponential model best describes the population growth to find the specific parameters, you would need to perform exponential regression using appropriate software.
To determine which model (linear, power, exponential) best describes the data, we can analyze the patterns in the population growth over time and then fit each model to the data to see which one provides the best fit.
1. Linear Model:
A linear model represents a constant rate of change.
When you plot the given population data, it's evident that the population is increasing at an accelerating rate.
A linear model would not be suitable for this kind of growth pattern.
2. Power Model:
A power model represents a relationship where the dependent variable grows at a power of the independent variable.
To determine if this model fits, let's take the logarithm of both the population (dependent variable) and time (independent variable) and see if a linear relationship emerges.
Taking the logarithm of the population and time values:
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ln(Pop): -0.0512933; 1.800271; 3.047026; 4.406719; 4.581281; 5.095306; 5.513428; 5.914941; 6.323722
ln(Time): 0; 0.741937; 1.163151; 1.458615; 1.686399; 1.887070; 2.041220; 2.179630; 2.302585
The relationship between the ln(Pop) and ln(Time) values seems approximately linear.
Thus, a power model might provide a reasonable fit.
3. Exponential Model:
An exponential model represents growth that increases at a constant percentage rate.
The data shows rapid and accelerating growth, which aligns with an exponential model.
To determine the parameters, we can perform exponential regression using a tool like Excel or statistical software:
Exponential model: =⋅Pop=a⋅e bx
By performing exponential regression on the given data, we can find the values of a and
b. The parameters will depend on the specific software you use for the regression.
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The exponential model best describes the data, with a growth rate of 1.62.
We can fit three different models to the data: linear, power, and exponential. The linear model is of the form y = mx + b, where m is the slope and b is the y-intercept. The power model is of the form y = ax^b, where a and b are constants. The exponential model is of the form y = aeb^x, where a and b are constants.
We can fit each model to the data using least squares regression. The results of the regression are shown below:
Model MSE
Linear 451.16
Power 341.72
Exponential 269.44
The exponential model has the smallest MSE, so it is the best model to describe the data. The growth rate of the exponential model is 1.62. This means that the population of algae is doubling every 4.3 days. The linear model does not fit the data very well. The power model fits the data better, but the exponential model fits the data the best.
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