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 number of years from today.
\[
p(t)=1500(1.019)^{t}
\]
Find the initial price of the item.
s]
Does the function represent growth or decay?
growth
decay
By what percent does the price change each year?
$\square \%$
Answer
Final Answer: The initial price of the item is \(\boxed{1500}\) dollars. The function represents growth. The price changes by approximately \(\boxed{1.9\%}\) each year.
Step 1 ：The initial price of the item is the price at time t=0. We can find this by substituting t=0 into the given function: \(p(0)=1500(1.019)^{0}\).
Step 2 ：The function represents growth if the base of the exponent (1.019) is greater than 1, and decay if it is less than 1. In this case, the base is greater than 1, so the function represents growth.
Step 3 ：The percent change each year is represented by the base of the exponent minus 1, multiplied by 100. So, \((1.019-1)*100\) gives the percent change each year.
Step 4 ：Final Answer: The initial price of the item is \(\boxed{1500}\) dollars. The function represents growth. The price changes by approximately \(\boxed{1.9\%}\) each year.
