We appreciate your visit to I bought a new car for tex 20 000 tex It is worth only tex 90 tex of its value each year Write an exponential. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
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.
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