Questions: A population of beetles are growing according to a linear growth model. The initial population (week 0) is P0=5, and the population after 6 weeks is P6=47. Find an explicit formula for the beetle population after n weeks. Pn= After how many weeks will the beetle population reach 131? weeks

Solution Steps

To find the explicit formula for the beetle population after \( n \) weeks, we need to determine the linear growth rate. Given the initial population and the population after 6 weeks, we can calculate the weekly increase. Once we have the growth rate, we can formulate the explicit equation. To find the number of weeks it takes for the population to reach 131, we can solve the equation for \( n \).

Given the initial population \( P_0 = 5 \) and the population after 6 weeks \( P_6 = 47 \), we can calculate the growth rate \( r \) as follows:

\[ r = \frac{P_6 - P_0}{6} = \frac{47 - 5}{6} = \frac{42}{6} = 7 \]

The explicit formula for the beetle population after \( n \) weeks can be expressed as:

\[ P_n = P_0 + r \cdot n = 5 + 7n \]

To find the number of weeks \( n \) required for the population to reach \( 131 \), we set up the equation:

\[ 131 = 5 + 7n \]

Solving for \( n \):

\[ 131 - 5 = 7n \implies 126 = 7n \implies n = \frac{126}{7} = 18 \]

Final Answer