To solve this problem, we need to derive the explicit formula for the beetle population after [tex]\( n \)[/tex] weeks based on the given initial population and the population after 8 weeks. Then, we'll use this formula to determine the number of weeks required for the population to reach 93.
1. Step 1: Determine the Linear Growth Rate
The initial population at week 0 is [tex]\( P_0 = 9 \)[/tex].
The population at week 8 is [tex]\( P_8 = 33 \)[/tex].
Since the population is growing linearly, we can calculate the weekly growth rate using the formula for the slope of a line:
[tex]\[
\text{Growth rate} = \frac{P_8 - P_0}{8}
\][/tex]
Plugging in the given values:
[tex]\[
\text{Growth rate} = \frac{33 - 9}{8} = \frac{24}{8} = 3
\][/tex]
2. Step 2: Derive the Explicit Formula
The population grows by a constant amount each week, so 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]
Substituting [tex]\( P_0 = 9 \)[/tex] and the growth rate of 3:
[tex]\[
P_n = 9 + 3n
\][/tex]
Therefore, the explicit formula for the beetle population after [tex]\( n \)[/tex] weeks is:
[tex]\[
P_n = 9 + 3n
\][/tex]
3. Step 3: Determine When the Population Reaches 93
We need to find the number of weeks [tex]\( n \)[/tex] when the population [tex]\( P_n \)[/tex] is 93:
[tex]\[
93 = 9 + 3n
\][/tex]
Subtract 9 from both sides to isolate the term involving [tex]\( n \)[/tex]:
[tex]\[
93 - 9 = 3n
\][/tex]
[tex]\[
84 = 3n
\][/tex]
Divide both sides by 3 to solve for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{84}{3} = 28
\][/tex]
Thus, the beetle population will reach 93 after [tex]\( 28 \)[/tex] weeks.
