Question

How many pennies do you save on the twenty-third day?

Answer 1

Given:

`\begin{gathered} Day1=1penny \\ Day2=2pennies \\ Day3=4pennies \\ Day4=8pennies \end{gathered}`

An exponential function is of the form:

`y=ab^x`

To get the exponential function for the relation;

`\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;} \\ (2)/(1)=(ab^2)/(ab) \\ 2=b \\ b=2 \\ \\ Substituting\text{ b into equation \lparen1\rparen;} \\ ab=1 \\ a(2)=1 \\ 2a=1 \\ a=(1)/(2) \\ \\ \\ \\ Hence,\text{ the function is;} \\ y=(1)/(2)(2^x) \end{gathered}`

Part A:

Relating this to the parameters given:

The exponential function that models the problem is;

`f(t)=(1)/(2)(2^t)`

Part B:

On the twenty-third day,

`\begin{gathered} when\text{ }t=23,\text{ the pennies saved will be;} \\ \\ f(t)=(1)/(2)(2^(23)) \\ f(t)=(2^(23))/(2) \\ f(t)=4,194,304\text{ pennies} \end{gathered}`

Therefore, he would have saved 4,194,304 pennies on the twenty-third day.