Mike had $13,500 in his checkings account at the beginning of year 2021. Each year, the net value of his account increases by $8,400. If the value of the checkings account \(t\) years after 2021 was between $33,500 and $43,000, which of the following must be true?

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The amount \(A\) in the checkings account \(t\) years after 2021 can be modeled by a simple linear equation:

$$ A=8{,}400t+13{,}500 $$

We are trying to find the value of \(t\) when the amount is between \(33{,}500\) and \(43{,}000\).

$$ 33{,}500 \lt A \lt 43{,}000 $$ $$ 33{,}500 \lt 8{,}400t+13{,}500 \lt 43{,}000$$ $$ 20{,}000 \lt 8{,}400t \lt 26{,}500 $$ $$ 2.38 \lt t \lt 3.51 $$

The range \( \boxed{2 \lt t \lt 4} \) includes all values from our inequality above.

We can calculate the amount in the checkings account in the first few years.

$$ \begin{array}{|c|c|} \hline \text{$t$} & \text{amount} \\ \hline 0 & \$13{,}500 \\ \hline 1 & \$21{,}900 \\ \hline 2 & \$30{,}300 \\ \hline 3 & \$38{,}700 \\ \hline 4 & \$47{,}100 \\ \hline \end{array} $$

The given amounts in our checkings account are $33,500 and $43,000, which are between \(t=2\) and \(t=4\).