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 9 weeks is P0=113. Find an explicit formula for the beetle population after n weeks. Pn= After how many weeks will the beetle population reach 269? weeks

A population of beetles are growing according to a linear growth model, The initial population (week 0) is P0=5, and the population after 9 weeks is P0=113.

Find an explicit formula for the beetle population after n weeks.
Pn=

After how many weeks will the beetle population reach 269?
weeks

Transcript text: A population of beetles are growing according to a linear growth model, The initial population (week 0) is $P_{0}=5$, and the population after 9 weeks is $P_{0}=113$. Find an explicit formula for the beetle population after $n$ weeks. \[ P_{\mathrm{n}}= \] After how many weeks will the beetle population reach 269? weeks

Solution

Solution Steps

Step 1: Calculate the Rate of Growth (r)

To find the rate of growth per week, we use the formula $r = \frac{P_t - P_0}{t}$. Given that $P_0 = 5$, $P_t = 113$, and $t = 9$ weeks, we calculate $r$ as follows: \[r = \frac{113 - 5}{9} = 12\]

Step 2: Calculate the Population After n Weeks (P_n)

Using the rate of growth $r$ and the initial population $P_0$, we can find the population after $n$ weeks using the formula $P_n = P_0 + rn$. Substituting $P_0 = 5$, $r = 12$, and $n = 269$, we get: \[P_n = 5 + 12 \times 269 = 3233\]