\begin{tabular}{l} Suppose that the future price \( p(t) \) of a certain item is given by the following exponential function. In this function, \( p(t) \) is \\ measured in dollars and \( t \) is the number of years from today. \\ \( p(t)=800(1.041)^{t} \) \\ Find the initial price of the item. \\ \( \$ \prod \) \\ \hline \( \begin{array}{l}\text { Does the function represent growth or decay? } \\ \text { Browth what percent does the price change each year? } \\ \square \%\end{array} \) \\ \hline\end{tabular}
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The initial price of the item can be found by evaluating the function at time \( t = 0 \). Plugging \( t = 0 \) into the equation \( p(t) = 800(1.041)^{t} \) gives us \( p(0) = 800(1.041)^{0} = 800 \cdot 1 = 800 \). Therefore, the initial price of the item is **\$800**. This function represents growth because the base of the exponent, \( 1.041 \), is greater than 1. To find the growth percentage, we can take \( (1.041 - 1) \times 100 \), which equals \( 4.1\%\). So, the price increases by **4.1%** each year.
