To determine the correct equation that represents the scenario where Chandler employs 30 people and increases the number of seasonal workers by 5% each week, we need to recognize that this is a situation involving exponential growth.
### Key Concepts:
1. Initial Number of Employees: Chandler initially employs 30 people.
2. Weekly Increase Rate: The number of employees grows by [tex]\(5\%\)[/tex] each week.
### Understanding Exponential Growth:
Exponential growth can be described by the equation:
[tex]\[ f(x) = a \cdot (1 + r)^x \][/tex]
where:
- [tex]\( f(x) \)[/tex] is the number of employees after [tex]\(x\)[/tex] weeks,
- [tex]\( a \)[/tex] is the initial number of employees,
- [tex]\( r \)[/tex] is the growth rate per period (in this case, per week),
- [tex]\( x \)[/tex] is the number of periods (in this case, weeks).
### Applying the Given Data:
- Initial Number of Employees ([tex]\( a \)[/tex]): 30
- Growth Rate ([tex]\( r \)[/tex]): 5% or [tex]\(0.05\)[/tex]
Plugging in these values into the exponential growth formula:
[tex]\[ f(x) = 30 \cdot (1 + 0.05)^x \][/tex]
### Simplification:
Simplifying inside the parentheses:
[tex]\[ f(x) = 30 \cdot (1.05)^x \][/tex]
### Result:
So, the correct equation that represents this scenario is:
[tex]\[ f(x) = 30(1.05)^x \][/tex]
Now let’s evaluate the incorrect options:
1. [tex]\( f(x)=30(0.05)^x \)[/tex]:
- This incorrectly uses [tex]\(0.05\)[/tex] as the base, which implies a constant decrease and not a 5% increase.
2. [tex]\( f(x)=30+1.05x \)[/tex]:
- This is a linear equation, implying a constant increase of 1.05 each week, which doesn't reflect the compounding nature of the problem.
3. [tex]\( f(x)=30+0.05x \)[/tex]:
- This is also a linear equation, implying a constant increase of 0.05 employees per week, which is incorrect.
Therefore, the correct equation that models Chandler's scenario of increasing his number of seasonal workers by 5% each week is:
[tex]\[ f(x)=30(1.05)^x \][/tex]
