\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}

Question

\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.

Suppose that the future price of a certain item is given by the following exponential function. In this function, is

measured in dollars and is the number of years from today.

Find the initial price of the item.

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Suppose that the future price of a certain item is given by the following exponential function. In this function, is measured in dollars and is the number of years from today. Find the initial price of the item.