Question
Georgia has a ring valued at $100. It decreases 2% in value each year. Is this an example of a linear or exponential relationship.
Equation:
f(x)=0.5x 60 50
f(x)=1.5x 40
f(x)=2x+100 30
f(x)=100x 20 10
f(x)=100(0.5)^x 0
f(x)=100(0.98)^x
f(x)=100(1.5)^x
f(x)=100(2)^x
Equation:
f(x)=0.5x 60 50
f(x)=1.5x 40
f(x)=2x+100 30
f(x)=100x 20 10
f(x)=100(0.5)^x 0
f(x)=100(0.98)^x
f(x)=100(1.5)^x
f(x)=100(2)^x
Answers
GPT-4o mini
The decrease in value of the ring can be modeled as an exponential relationship because the value is decreasing by a consistent percentage (2%) each year rather than a fixed amount.
When valuing the ring, its initial value is $100, and it decreases by 2% each year, which can be modeled with the equation:
\[ f(x) = 100(0.98)^x \]
In this equation, \( x \) represents the number of years, and \( 0.98 \) is \( 1 - 0.02 \) (since it loses 2% of its value each year), indicating that the ring retains 98% of its value each year.
### Summary:
- Relationship Type: Exponential
- Correct Equation: \( f(x) = 100(0.98)^x \)
When valuing the ring, its initial value is $100, and it decreases by 2% each year, which can be modeled with the equation:
\[ f(x) = 100(0.98)^x \]
In this equation, \( x \) represents the number of years, and \( 0.98 \) is \( 1 - 0.02 \) (since it loses 2% of its value each year), indicating that the ring retains 98% of its value each year.
### Summary:
- Relationship Type: Exponential
- Correct Equation: \( f(x) = 100(0.98)^x \)