\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 $$ \begin{array}{l} ext { Does the function represent growth or decay? } \ ext { Browth what percent does the price change each year? } \ \square \%\end{array} $$ \ \hline\end{tabular}

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\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 $$ \begin{array}{l}	ext { Does the function represent growth or decay? } \ 	ext { Browth what percent does the price change each year? } \ \square \%\end{array} $$ \ \hline\end{tabular}

UpStudy AI · Accepted Answer

**Step 1. Find the initial price**

To find the initial price, substitute $$ t = 0 $$ into the function:
$$
p(0) = 800(1.041)^0
$$
Since $$ (1.041)^0 = 1 $$, we have:
$$
p(0) = 800 \times 1 = 800
$$

**Step 2. Determine if the function represents growth or decay**

The function is given by:
$$
p(t) = 800(1.041)^t
$$
Since the base $$ 1.041 $$ is greater than 1, the function represents **growth**.

**Step 3. Calculate the percent change each year**

The factor $$ 1.041 $$ indicates that the price is multiplied by $$ 1.041 $$ each year. The annual percentage increase is:
$$
(1.041 - 1) \times 100\% = 0.041 \times 100\% = 4.1\%
$$

Thus, the annual growth rate is **4.1%**.
The initial price is \$800. The function represents **growth**, with the price increasing by **4.1%** each year.

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