Solution:
Given:
[tex]\begin{gathered} Day1=1penny \\ Day2=2pennies \\ Day3=4pennies \\ Day4=8pennies \end{gathered}[/tex]An exponential function is of the form:
[tex]y=ab^x[/tex]To get the exponential function for the relation;
[tex]\begin{gathered} For\text{ day 1:} \\ 1=ab^1.........................(1) \\ For\text{ day 2:} \\ 2=ab^2.........................(2) \\ For\text{ day 3:} \\ 4=ab^3 \\ \\ \\ \\ Hence,\text{ equation \lparen2\rparen divided by equation \lparen1\rparen;} \\ \frac{2}{1}=\frac{ab^2}{ab} \\ 2=b \\ b=2 \\ \\ Substituting\text{ b into equation \lparen1\rparen;} \\ ab=1 \\ a(2)=1 \\ 2a=1 \\ a=\frac{1}{2} \\ \\ \\ \\ Hence,\text{ the function is;} \\ y=\frac{1}{2}(2^x) \end{gathered}[/tex]Part A:
Relating this to the parameters given:
The exponential function that models the problem is;
[tex]f(t)=\frac{1}{2}(2^t)[/tex]Part B:
On the twenty-third day,
[tex]\begin{gathered} when\text{ }t=23,\text{ the pennies saved will be;} \\ \\ f(t)=\frac{1}{2}(2^{23}) \\ f(t)=\frac{2^{23}}{2} \\ f(t)=4,194,304\text{ pennies} \end{gathered}[/tex]Therefore, he would have saved 4,194,304 pennies on the twenty-third day.