Sure, let's go through each step in detail:
### Step 1:
Identify the initial amount [tex]\( a \)[/tex] and the growth rate [tex]\( r \)[/tex]
The initial amount [tex]\( a \)[/tex] is [tex]$45,000 and the depreciation rate \( r \) is 20%.
### Step 2:
Use the exponential decay formula \( y = a(1 - r)^t \)
Given:
- \( a = 45,000 \)
- \( r = 0.2 \)
The exponential decay model will be:
\[ y = 45,000 \times (1 - 0.2)^t \]
\[ y = 45,000 \times (0.8)^t \]
### Step 3:
Identify the value of \( t \) and use it in the exponential decay model to solve the problem
We want to find the value of the boat after 4 years. So, \( t = 4 \).
Substitute \( t = 4 \) into the model:
\[ y = 45,000 \times (0.8)^4 \]
From the given data:
\[ y \approx 18,432.00 \]
So, the value of the boat after 4 years is approximately $[/tex]18,432.00.
### Step 4:
Create a table of values and graph the function
Let's create a table of values for the first 5 years:
[tex]\[
\begin{array}{|c|c|}
\hline
t \text{ (years)} & y \text{ (value of the boat)} \\
\hline
0 & 45,000.00 \\
1 & 36,000.00 \\
2 & 28,800.00 \\
3 & 23,040.00 \\
4 & 18,432.00 \\
5 & 14,745.60 \\
\hline
\end{array}
\][/tex]
### Verification with Graphing:
You can graph the exponential decay function [tex]\( y = 45,000 \times (0.8)^t \)[/tex] using graphing software or a graphing calculator. The table of values above can help you to plot the points: (0, 45000), (1, 36000), (2, 28800), (3, 23040), (4, 18432), and (5, 14745.6).
The graph will show a decreasing curve, starting at $45,000 and approaching zero as [tex]\( t \)[/tex] increases.
