Maria purchased a new pickup truck that is valued at $25,000. The value of the truck decreases at a rate of 20% each year. What will Maria’s truck be worth after 7 years?

Question

Because the value of the truck decreases by a given percentage each year, this situation can be modeled by an exponential______________________ function.

The standard form of an exponential decay function is f(x)=___(1-___)^x.

The function that models this situation is f(x)=___(___)^x.

Answer 1

To analyze the depreciation of Maria’s truck, we can use the concept of exponential decay.

1. 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 \(1 - 0.20 = 0.80\).

2. 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.

3. 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: \[ f(x) = 25000(1 - 0.20)^x = 25000(0.80)^x. \]

4. Modeling the Situation: Hence, the function that models this situation is: \[ f(x) = 25000(0.80)^x. \]

5. 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 \(f(x) = 25000(0.80)^x\).