To find an exponential function that models the situation described, we start by observing the values given in the table:
- At \( x = 0 \), \( v(0) = 12000 \)
- At \( x = 1 \), \( v(1) = 10800 \)
- At \( x = 2 \), \( v(2) = 9720 \)
- At \( x = 3 \), \( v(3) = 8748 \)
In an exponential decay scenario, the value of the motorcycle can typically be modeled by the function:
\[ v(x) = v_0 \cdot a^x \]
where:
- \( v_0 \) is the initial value (at \( x = 0 \)),
- \( a \) is the decay factor.
From the first data point, we can see that \( v_0 = 12000 \).
Next, we need to calculate the decay factor \( a \). To find \( a \), we can use the second data point:
\[ v(1) = 12000 \cdot a \] Setting this equal to the actual value at \( x = 1 \):
\[ 10800 = 12000 \cdot a \]
To solve for \( a \):
\[ a = \frac{10800}{12000} = \frac{9}{10} = 0.9 \]
Next, we can check whether this decay factor holds for the other years:
For \( x = 2 \): \[ v(2) = 12000 \cdot (0.9)^2 = 12000 \cdot 0.81 = 9720 \quad \text{(correct)} \]
For \( x = 3 \): \[ v(3) = 12000 \cdot (0.9)^3 = 12000 \cdot 0.729 = 8748 \quad \text{(correct)} \]
Thus, the exponential function that models this situation is:
\[ v(x) = 12000 \cdot (0.9)^x \]
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