Let's break down the solution to the given problem step-by-step:
### 1. Finding the Initial Price of the Item
The initial price of the item can be found by evaluating the exponential function [tex]\( p(t) \)[/tex] at [tex]\( t = 0 \)[/tex].
Given the function:
[tex]\[ p(t) = 1500 \times (1.025)^t \][/tex]
Plugging in [tex]\( t = 0 \)[/tex]:
[tex]\[
p(0) = 1500 \times (1.025)^0
\][/tex]
We know that any number raised to the power of 0 is 1, so:
[tex]\[
(1.025)^0 = 1
\][/tex]
Therefore:
[tex]\[
p(0) = 1500 \times 1 = 1500
\][/tex]
Initial Price of the Item: $1500.00
### 2. Determining if the Function Represents Growth or Decay
To determine whether the function represents growth or decay, we need to look at the base of the exponential function, which is 1.025 in this case.
- If the base is greater than 1, the function represents growth.
- If the base is between 0 and 1, the function represents decay.
Since 1.025 is greater than 1, the function represents growth.
The function represents: Growth
### 3. Calculating the Annual Percent Change in Price
To find the annual percentage change in price, we use the growth rate given by the base of the exponential function, which is 1.025.
The formula to convert a growth rate into a percentage change is:
[tex]\[
\text{Percentage Change} = ( \text{Growth Rate} - 1 ) \times 100
\][/tex]
So here, the growth rate is 1.025:
[tex]\[
\text{Percentage Change} = (1.025 - 1) \times 100
\][/tex]
Calculating this:
[tex]\[
\text{Percentage Change} = 0.025 \times 100 = 2.5
\][/tex]
The annual percentage change in price: 2.5%
