To determine which exponential function models the situation, we need to analyze the values provided for the motorcycle after each number of years. The values provided in the table indicate a depreciation of the motorcycle's value.
The values provided are:
- After 0 years: 12,000 (initial value)
- After 1 year: 10,800
- After 2 years: 9,720
- After 3 years: 8,748
To find the correct exponential function, we can notice how much the value decreases over time.
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The value after 1 year is 10,800, which is a decrease from 12,000. \[ \frac{10800}{12000} = 0.9 \text{ (or a 10% decrease)} \]
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The value after 2 years of depreciation from 10,800: \[ \frac{9720}{10800} = 0.9 \text{ (another 10% decrease)} \]
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The value after 3 years from 9,720: \[ \frac{8748}{9720} = 0.9 \text{ (again, a 10% decrease)} \]
The value is consistently multiplying by 0.9 each year. Therefore, we can conclude that the motorcycle depreciates by 10% each year. The model for the value after \(x\) years would be given by the initial value multiplied by a decay factor.
This leads us to the following exponential function: \[ v(x) = 12000 \times (0.9)^x \]
From the options provided, the correct model is: "v(x) = 12,000 * (0.9)^x."
1 answer