Question

The population of a certain insects is initially 312 insect population shows the exponential growth of 5% each week right the exponential equation to model the insect population why after X weeks

Answer 1

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:

\[
P(x) = 312 \cdot (1.05)^x
\]