To find the correct exponential equation for the given points (0, 30000) and (1, 22500), we can start by identifying the general form of an exponential decay equation:
\[ y = a \cdot b^x \]
Where:
- \( a \) is the initial value (the value of the car at \( x = 0 \)).
- \( b \) is the decay factor per year.
- \( x \) is the number of years since purchase.
From the points provided:
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At \( x = 0 \), the value of the car is \( y = 30000 \). This gives us \( a = 30000 \). So, our equation becomes: \[ y = 30000 \cdot b^x \]
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At \( x = 1 \), the value of the car is \( y = 22500 \). Substituting \(x = 1\) and \(y = 22500\) into the equation: \[ 22500 = 30000 \cdot b^1 \]
To solve for \(b\), we can rearrange the equation: \[ b = \frac{22500}{30000} = \frac{3}{4} \]
Now, we can substitute both \( a \) and \( b \) back into the equation: \[ y = 30000 \cdot \left(\frac{3}{4}\right)^x \]
Now, examining the response options:
- \( y=0.75(30,000)x \): This is incorrect because it doesn't follow the exponential form.
- \( y=30,000\left(\frac{3}{4}\right)^x \): This matches our derived equation.
- \( y=30,000\left(\frac{4}{3}\right)^x \): This is incorrect as it suggests an increase rather than decay.
- \( y=22,500\left(\frac{3}{4}\right)^x \): This is incorrect for the same reason—while it uses the decay factor, it incorrectly uses a different initial value.
Thus, the correct option is: \[ y=30,000\left(\frac{3}{4}\right)^x \]