To model the population of the insects using an exponential growth equation, we can use the general formula for exponential growth:
\[ P(t) = P_0 \cdot e^{rt} \]
Where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the growth rate (expressed as a decimal),
- \( t \) is the time in weeks,
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
However, since we have a weekly growth rate and it is a 5% increase, we can use a simpler formula that directly calculates the population over discrete time intervals (like weeks):
\[ P(t) = P_0 \cdot (1 + r)^t \]
Where \( r \) is the growth rate in decimal form (\( r = 0.05 \) for 5%).
Given the initial population \( P_0 = 312 \), the equation to model the insect population \( P \) after \( x \) weeks is:
\[ P(x) = 312 \cdot (1 + 0.05)^x \]
This can be simplified to:
\[ P(x) = 312 \cdot (1.05)^x \]
So, the exponential equation to model the insect population after \( x \) weeks is:
\[ P(x) = 312 \cdot (1.05)^x \]