To model the population of the insects after \( x \) weeks with an initial population of 312 and a growth rate of 5% per week, you can use the exponential growth formula:
\[ y = y_0 \cdot (1 + r)^x \]
where:
- \( y_0 \) is the initial population,
- \( r \) is the growth rate (as a decimal),
- \( x \) is the number of weeks.
In this case:
- \( y_0 = 312 \)
- \( r = 0.05 \)
Plugging these values into the equation gives:
\[ y = 312 \cdot (1 + 0.05)^x \]
This simplifies to:
\[ y = 312 \cdot (1.05)^x \]
So, the exponential equation to model the insect population after \( x \) weeks is:
\[ y = 312 \cdot (1.05)^x \]
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