Answer :
To solve the problem of finding an exponential function that represents the value of the car after [tex]\( x \)[/tex] years, we need to consider the following details:
1. Initial Value: The car's initial value is [tex]$20,000.
2. Depreciation Rate: Each year, the car is worth 90% of its previous value. This means the car loses 10% of its value every year. In mathematical terms, 90% of something is the same as multiplying by 0.90.
3. Exponential Decay Function: The problem involves an exponential decay situation because the car's value decreases by a fixed percentage each year. The general form of an exponential function for depreciation is:
\[
f(x) = a \cdot (b)^x
\]
where:
- \( a \) is the initial value,
- \( b \) is the decay factor (in this case, 0.90 since the car retains 90% of its value each year),
- \( x \) is the number of years.
For this problem:
- The initial value \( a \) is $[/tex]20,000.
- The decay factor [tex]\( b \)[/tex] is 0.90.
So, the exponential function representing the value of the car after [tex]\( x \)[/tex] years is:
[tex]\[
f(x) = 20000 \cdot (0.90)^x
\][/tex]
This function accurately describes how the car's value decreases over time, reflecting the 90% retention (or 10% loss) in value each year.
1. Initial Value: The car's initial value is [tex]$20,000.
2. Depreciation Rate: Each year, the car is worth 90% of its previous value. This means the car loses 10% of its value every year. In mathematical terms, 90% of something is the same as multiplying by 0.90.
3. Exponential Decay Function: The problem involves an exponential decay situation because the car's value decreases by a fixed percentage each year. The general form of an exponential function for depreciation is:
\[
f(x) = a \cdot (b)^x
\]
where:
- \( a \) is the initial value,
- \( b \) is the decay factor (in this case, 0.90 since the car retains 90% of its value each year),
- \( x \) is the number of years.
For this problem:
- The initial value \( a \) is $[/tex]20,000.
- The decay factor [tex]\( b \)[/tex] is 0.90.
So, the exponential function representing the value of the car after [tex]\( x \)[/tex] years is:
[tex]\[
f(x) = 20000 \cdot (0.90)^x
\][/tex]
This function accurately describes how the car's value decreases over time, reflecting the 90% retention (or 10% loss) in value each year.
