Question

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?

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