Answer: [tex]f(x)=\text{ 4000\lparen}\frac{\text{ }x+\text{ 100}}{100})^{x\text{ +2}}\text{ }[/tex]
Explanation:
Amount invested by Katelyn = $4000
rate = x%
time = x + 2 years
n = number of times compounded
n = annually = 1
We need to find the function that models the total amount
To find the function, we will apply the compound interest formula:
[tex]\begin{gathered} $FV\text{ = P(1 +}\frac{r}{n})^{nt}$ \\ \\ P\text{ = 4000} \\ t\text{ = x + 2} \\ n\text{ = 1} \\ \text{ r = x\% = x/100} \\ FV=\text{ total amount } \end{gathered}[/tex]
substitute the values into the formula:
[tex]\begin{gathered} FV=\text{ 4000\lparen 1 + }\frac{\frac{x}{100}}{1})^{1\times(x\text{ + 2\rparen}} \\ \\ FV=\text{ 4000\lparen1 + }\frac{x}{100})^{x+2} \end{gathered}[/tex][tex]FV=\text{ 4,000\lparen}\frac{100+\text{ }x}{100})^{x\text{ +2}}\text{ }[/tex]
let the function that represents the total amount = f(x)
[tex]f(x)=\text{ 4000\lparen}\frac{\text{ }x+\text{ 100}}{100})^{x\text{ +2}}\text{ }[/tex]
