To model the insect population using an exponential growth equation, we can start with the general form of the exponential growth formula:
\[ y = y_0 \cdot (1 + r)^x \]
where:
- \( y \) is the population after \( x \) weeks,
- \( y_0 \) is the initial population,
- \( r \) is the growth rate (as a decimal),
- \( x \) is the number of time periods (weeks in this case).
In your case:
- The initial population \( y_0 = 312 \),
- The growth rate \( r = 5% = 0.05 \).
Plugging in these values, the equation becomes:
\[ y = 312 \cdot (1 + 0.05)^x \]
which simplifies to:
\[ y = 312 \cdot (1.05)^x \]
Therefore, the exponential equation to model the insect population after \( x \) weeks is:
\[ y = 312 \cdot (1.05)^x \]
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