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The exponential decay model is a mathematical way to represent how things lose value over time. In this context, it describes how a car's trade-in value decreases each year. Think of exponential decay as a gradual decline where the amount decreases by a consistent percentage every time period, in this case, every year.
Here, the factor that influences the decrease is the base of the exponent, which is 0.85. This number tells us by what percentage the value of the car keeps each year, which in this scenario is 85% of its value from the previous year.
So, if the car's value was decreasing following this model, it means the car keeps 85% and loses 15% of its remaining value annually. This kind of decay is common in processes like depreciation, which affects assets like vehicles or electronic goods. It captures both the idea of losing value and the rate at which this happens.

The trade-in value formula is a simple but powerful tool to calculate how much a car is worth over time. Essentially, it combines the concepts of depreciation and initial cost into one equation. The formula used here is:

• \( V(t) = 0.78 C (0.85)^{t-1} \)

Where:

• \( V(t) \) is the trade-in value of the car after \( t \) years
• \( C \) is the initial cost of the car; here, \( C \) is 25,000 dollars
• \( 0.85 \) is the decay factor showing the percentage of value retained each year
• \( t \) is the number of years we are looking at

The component \( 0.78 \) accounts for an additional immediate depreciation—or simply, the starting trade-in value as a percentage of the purchase price. Using this formula allows you to estimate the car's worth effortlessly, giving you a clear idea of potential resale value over time.

Breaking down problems into smaller, manageable tasks can make solving them easier to tackle, as seen in this exercise. Here's how you can calculate trade-in values using the step-by-step approach:

Start with understanding the formula and plugging in known values. Begin by substituting the initial purchase price (\( C \)) and the specific number of years (\( t \)) into the trade-in value formula. Let's illustrate these steps.

For 1 year, the steps would go:

• Substitute \( C = 25000 \) and \( t = 1 \) into \( V(t) = 0.78 \times 25000 \times (0.85)^{0} \)
• Calculate \( V(1) = 0.78 \times 25000 \)
• The result is \( 19500 \) dollars.

For 4 years:

• Substitute \( C = 25000 \) and \( t = 4 \)
• Compute \( (0.85)^3 \), utilizing a calculator gives you approximately \(0.614125\).
• Calculate \( V(4) = 0.78 \times 25000 \times 0.614125 \approx 11997 \)

For 7 years, follow similar steps:

• Substitute \( C = 25000 \) and \( t = 7 \)
• Find \( (0.85)^6 \approx 0.377149 \)
• Then, \( V(7) = 0.78 \times 25000 \times 0.377149 \approx 7373 \)

This approach ensures even complex-seeming calculations remain accessible and straightforward.