Math Problem Statement
A car was valued at $39,000 in the year 1995. The value depreciated to $14,000 by the year 2003.
A) What was the annual rate of change between 1995 and 2003? r = Round the rate of decrease to 4 decimal places.
Solution
To find the annual rate of change, we can assume that the value of the car depreciated exponentially. The formula for exponential decay is:
Where:
- is the value after years,
- is the initial value,
- is the annual rate of decrease,
- is the number of years.
Given:
- Initial value (in 1995),
- Final value (in 2003),
- years.
Now we can solve for by using the formula:
First, isolate by dividing both sides by 39,000:
Now, solve for , and then for . I'll calculate this for you.The annual rate of decrease between 1995 and 2003 is approximately 0.1202, or 12.02%.
Would you like further details on this process, or do you have any other questions?
Here are five related questions to consider:
- How does the formula for exponential decay differ from linear depreciation?
- What would the car's value be in 2010 if the same rate of depreciation continued?
- How do you calculate depreciation using straight-line methods versus exponential?
- If the rate of decrease was 15%, what would the car's value have been in 2003?
- How can you calculate the rate of depreciation if given only two time points?
Tip: For exponential depreciation, the rate stays consistent over time, making it different from straight-line depreciation, which assumes a fixed amount lost each year.
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Math Problem Analysis
Mathematical Concepts
Exponential Decay
Depreciation
Algebra
Formulas
V(t) = V_0 * (1 - r)^t
Annual rate of change formula: (V(t)/V_0)^(1/t) = 1 - r
Theorems
Exponential Decay Formula
Suitable Grade Level
Grades 9-12
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