\begin{tabular}{l} How does $$ f(t)=1.15^{t} $$ change over the interval from $$ t=4 $$ to $$ t=5 $$ ? \ $$ f(t) $$ decreases by a factor of 1.15 \ $$ f(t) $$ decreases by 1.15 \ $$ f(t) $$ increases by $$ 115 \% $$ \ $$ f(t) $$ increases by a factor of 1.15 \ \hline\end{tabular}

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To determine how the function $$ f(t) = 1.15^{t} $$ changes as $$ t $$ increases from 4 to 5, let's analyze the behavior step-by-step.

1. **Calculate $$ f(4) $$ and $$ f(5) $$:**
   $$
   f(4) = 1.15^{4} \approx 1.7490
   $$
   $$
   f(5) = 1.15^{5} \approx 2.0114
   $$

2. **Determine the factor of change:**
   $$
   \text{Factor of Change} = \frac{f(5)}{f(4)} = \frac{1.15^{5}}{1.15^{4}} = 1.15^{5-4} = 1.15
   $$

3. **Interpret the Result:**
   - A factor of **1.15** means that $$ f(t) $$ **increases** by **15%** when $$ t $$ increases from 4 to 5.
   - Increasing by a factor of **1.15** is equivalent to multiplying the original value by **1.15**.

4. **Evaluate the Options:**
   - **Option 1:** $$ f(t) $$ decreases by a factor of 1.15 – *Incorrect* (the function increases).
   - **Option 2:** $$ f(t) $$ decreases by 1.15 – *Incorrect* (the function increases).
   - **Option 3:** $$ f(t) $$ increases by $$ 115\% $$ – *Incorrect* (an increase by 115% would mean multiplying by 2.15).
   - **Option 4:** $$ f(t) $$ increases by a factor of 1.15 – *Correct*.

**Answer:** $$ f(t) $$ increases by a factor of 1.15
$$ f(t) $$ increases by a factor of 1.15

How does change over the interval from to ?

decreases by a factor of 1.15

decreases by 1.15

increases by

increases by a factor of 1.15

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