To solve the problem step-by-step, let's break it down according to the information provided and the common methods used for analysis of exponential functions.
### Step 1: Determine the Initial Price
The initial price of the item is found by evaluating the function [tex]\( p(t) \)[/tex] when [tex]\( t = 0 \)[/tex].
Given the function:
[tex]\[ p(t) = 1500(1.019)^t \][/tex]
To find the initial price [tex]\( p(0) \)[/tex]:
[tex]\[ p(0) = 1500(1.019)^0 \][/tex]
Any number raised to the power of 0 is 1, therefore:
[tex]\[ p(0) = 1500 \cdot 1 = 1500 \][/tex]
So, the initial price of the item is:
[tex]\[ \$1500 \][/tex]
### Step 2: Determine if the Function Represents Growth or Decay
We identify whether the function represents growth or decay by looking at the base of the exponential expression, which is [tex]\( 1.019 \)[/tex].
If the base [tex]\( b \)[/tex] of the exponential function [tex]\( b > 1 \)[/tex], it indicates growth. Conversely, if [tex]\( b < 1 \)[/tex], it indicates decay. In this case, the base is [tex]\( 1.019 \)[/tex], which is greater than 1.
Therefore, the function represents:
[tex]\[ \text{growth} \][/tex]
### Step 3: Determine the Annual Percentage Change
To find the percentage change per year, we look at the growth factor which is [tex]\( 1.019 \)[/tex].
The formula for the percentage change based on the growth factor [tex]\( b \)[/tex] is:
[tex]\[ \text{Percentage Change} = (b - 1) \times 100\% \][/tex]
Substituting the growth factor [tex]\( 1.019 \)[/tex]:
[tex]\[ \text{Percentage Change} = (1.019 - 1) \times 100\% \][/tex]
[tex]\[ \text{Percentage Change} = 0.019 \times 100\% \][/tex]
[tex]\[ \text{Percentage Change} = 1.90\% \][/tex]
So, the price changes by:
[tex]\[ 1.90\% \text{ each year} \][/tex]
### Summary
1. The initial price of the item is [tex]\( \$1500 \)[/tex].
2. The function represents growth.
3. The price changes by 1.90\% each year.
