Kerri invested $38,976 into her first company. Her investment decreased by 3.5% each year over the first 3 years. Which exponential decay model could Kerri use to determine the amo... Kerri invested $38,976 into her first company. Her investment decreased by 3.5% each year over the first 3 years. Which exponential decay model could Kerri use to determine the amount her investment will be worth if it continues to decrease at the same rate?
Understand the Problem
The question is asking for the correct exponential decay model to represent Kerri's investment over time, given a specific annual rate of decrease. We need to identify the model that accurately reflects an investment that decreases by 3.5% each year.
Answer
The exponential decay model is given by $A(t) = A_0 e^{-0.0367t}$.
Answer for screen readers
The correct exponential decay model for Kerri's investment is $$ A(t) = A_0 e^{-0.0367t} $$
Steps to Solve
- Identify the general exponential decay formula
The general formula for exponential decay is given by $$ A(t) = A_0 e^{-kt} $$ where:
- $A(t)$ is the amount of the investment at time $t$,
- $A_0$ is the initial amount of the investment,
- $k$ is the decay constant,
- $t$ is the time in years,
- $e$ is the base of the natural logarithm (approximately 2.71828).
- Convert the percentage decrease to the decay constant
Given that Kerri's investment decreases by 3.5% each year, we can represent this as: $$ k = -\ln(1 - r) $$ where $r$ is the rate of decrease (in decimal form).
For a 3.5% decrease: $$ r = 0.035 $$
Therefore, $$ k = -\ln(1 - 0.035) $$
- Calculate the decay constant
Now calculate $k$: $$ k = -\ln(0.965) \approx 0.0367 $$
So, the decay constant $k$ represents the rate of decay per year.
- Write the specific exponential decay model
Now plugging $k$ back into our general decay formula, we have the specific model for Kerri's investment: $$ A(t) = A_0 e^{-0.0367t} $$
This is the equation that models Kerri's investment decreasing by 3.5% each year.
The correct exponential decay model for Kerri's investment is $$ A(t) = A_0 e^{-0.0367t} $$
More Information
Exponential decay models are commonly used in finance to represent depreciating investments or assets. In this model, the investment decreases continuously over time, reflecting a steady rate of decrease.
Tips
- A common mistake is to confuse exponential decay with linear decay. Exponential decay decreases at a rate proportional to the current value.
- Also, ensure that the percentage decrease is converted correctly to a decimal before calculations.
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