To analyze the depreciation of Maria’s truck, we can use the concept of exponential decay.
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Exponential Decay Function: The value of the truck decreases at a rate of 20% each year. This implies that it retains 80% of its value yearly, which can be represented by the decay factor \(1 - 0.20 = 0.80\).
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Standard Form: The standard form of an exponential decay function is typically expressed as: \[ f(x) = a(1 - r)^x \] where \(a\) is the initial value, \(r\) is the rate of decay, and \(x\) is the time in years.
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Values in the Function:
- The initial value \(a\) is $25,000.
- The decay rate \(r\) is 0.20.
Thus, reconfiguring the standard form gives us: \[ f(x) = 25000(1 - 0.20)^x = 25000(0.80)^x. \]
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Modeling the Situation: Hence, the function that models this situation is: \[ f(x) = 25000(0.80)^x. \]
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Calculating the Value After 7 Years: To find the value of the truck after 7 years, we substitute \(x = 7\) into the function: \[ f(7) = 25000(0.80)^7. \] First, we calculate \(0.80^7\): \[ 0.80^7 \approx 0.2097152. \] Now, multiplying by the initial value: \[ f(7) = 25000 \times 0.2097152 \approx 5242.88. \]
So, the truck will be worth approximately $5,242.88 after 7 years.
To summarize:
- The completed statements:
- This situation can be modeled by an exponential decay function.
- The standard form of an exponential decay function is \(f(x) = a(1 - r)^x\).
- The function that models this situation is \(f(x) = 25000(0.80)^x\).