By working with the exponential function, we will see that after 13.68 years the value of the card will be $27.09.
In how many years will the card be worth $27.09?
We know that the original value is $5.75, and it increases a 12% per year, then the value of the card is given by the exponential function:
[tex]f(x) = \$ 5.75*(1 + 0.12)^x[/tex]
We want to find the value of x such that:
f(x) = $27.09, so we need to solve:
[tex]f(x) = \$ 5.75*(1 + 0.12)^x = \$ 27.09\\\\(1.12)^x = 27.09/5.75[/tex]
Now we can apply the natural logarithm to both sides:
[tex]ln(1.12^x) = ln(27.09/5.75)\\\\x*ln(1.12) = ln(27.09/5.75)\\\\x = \frac{ln(27.09/5.75)}{ln(1.12)} = 13.68[/tex]
This means that after 13.68 years, the value of the card will be $27.09.
If you want to learn more about exponential functions:
https://brainly.com/question/11464095
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