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Write an exponential decay function to model this situation. Verify the solution by graphing the function.

1. A speedboat that cost [tex]$\$[/tex]45,000[tex]$ is depreciating at a rate of $[/tex]20\%[tex]$ per year. Find the value of the boat after 4 years.

\ \textless \ strong\ \textgreater \ Step 1:\ \textless \ /strong\ \textgreater \ Identify the initial amount $[/tex]a[tex]$ and the growth rate $[/tex]r[tex]$.
$[/tex]a = 45,000[tex]$ and $[/tex]r = 0.20[tex]$

\ \textless \ strong\ \textgreater \ Step 2:\ \textless \ /strong\ \textgreater \ Use $[/tex]y = a(1 - r)^t[tex]$ to write the exponential decay model (substitute values for $[/tex]a[tex]$ and $[/tex]r[tex]$).
$[/tex]y = 45,000(1 - 0.20)^t[tex]$

\ \textless \ strong\ \textgreater \ Step 3:\ \textless \ /strong\ \textgreater \ Identify the value of $[/tex]t[tex]$ and use it in the exponential decay model to solve the problem.
$[/tex]t = 4[tex]$

\ \textless \ strong\ \textgreater \ Step 4:\ \textless \ /strong\ \textgreater \ Create a table of values and graph the function.

\[
\begin{tabular}{|c|l|}
\hline
$[/tex]t[tex]$, years & $[/tex]y[tex]$, value of the boat \\
\hline
0 & $[/tex]45,000[tex]$ \\
\hline
1 & $[/tex]36,000[tex]$ \\
\hline
2 & $[/tex]28,800[tex]$ \\
\hline
3 & $[/tex]23,040[tex]$ \\
\hline
4 & $[/tex]18,432[tex]$ \\
\hline
5 & $[/tex]14,746[tex]$ \\
\hline
\end{tabular}
\]

\ \textless \ strong\ \textgreater \ Graph the function:\ \textless \ /strong\ \textgreater \

To graph the function, plot the values from the table on a coordinate plane, with $[/tex]t[tex]$ (years) on the x-axis and $[/tex]y$ (value of the boat) on the y-axis.


Sagot :

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.
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