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A population of beetles is growing according to a linear growth model. The initial population (week 0) is [tex]$P_0=5$[/tex], and the population after 8 weeks is [tex]$P_8=53$[/tex].

1. Find an explicit formula for the beetle population after [tex][tex]$n$[/tex][/tex] weeks.
[tex]P_n = \square[/tex]

2. After how many weeks will the beetle population reach 131?
[tex]\square[/tex] weeks


Sagot :

Sure, let's solve the problem step by step.

### Step 1: Calculate the growth rate

First, we need to determine the weekly growth rate of the beetle population. We know:
- At week 0, the population [tex]\( P_0 = 5 \)[/tex]
- At week 8, the population [tex]\( P_8 = 53 \)[/tex]

The growth over these 8 weeks is given by the difference between [tex]\( P_8 \)[/tex] and [tex]\( P_0 \)[/tex]:
[tex]\[ P_8 - P_0 = 53 - 5 = 48 \][/tex]

Since this growth occurred over 8 weeks, the growth rate per week is:
[tex]\[ \text{Growth rate} = \frac{P_8 - P_0}{8} = \frac{48}{8} = 6 \][/tex]

### Step 2: Construct the explicit formula

We want to find the population after [tex]\( n \)[/tex] weeks, [tex]\( P_n \)[/tex]. Given the linear growth model, the population increases by a constant amount each week, which we've calculated to be 6.

The general formula for the population after [tex]\( n \)[/tex] weeks can be written as:
[tex]\[ P_n = P_0 + (\text{growth rate}) \times n \][/tex]

Substitute [tex]\( P_0 \)[/tex] and the growth rate:
[tex]\[ P_n = 5 + 6n \][/tex]

### Step 3: Calculate the number of weeks to reach a population of 131

Now we need to determine after how many weeks the population will reach 131. We set [tex]\( P_n = 131 \)[/tex] and solve for [tex]\( n \)[/tex]:

[tex]\[ 131 = 5 + 6n \][/tex]

Subtract 5 from both sides:
[tex]\[ 126 = 6n \][/tex]

Divide by 6:
[tex]\[ n = \frac{126}{6} = 21 \][/tex]

### Summary

1. The explicit formula for the beetle population after [tex]\( n \)[/tex] weeks is:
[tex]\[ P_n = 5 + 6n \][/tex]

2. The beetle population will reach 131 after:
[tex]\[ 21 \text{ weeks} \][/tex]
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