A quantity with an initial value of 6200 decays continuously at a rate of 5.5% per month. What is the value of the quantity after 4 years, to the nearest hundredth?
To solve this problem, we can use the formula for exponential decay:
\[ A = P \cdot e^{rt} \]
where:
- A is the final amount
- P is the initial quantity
- e is the base of the natural logarithm (approximately equal to 2.71828)
- r is the decay rate per time period (in this case, per month)
- t is the time in months
Given:
- P = 6200
- r = 5.5% = 0.055
- t = 4 years = 4 * 12 = 48 months
Substitute these values into the formula:
\[ A = 6200 \cdot e^{0.055 \cdot 48} \]
\[ A = 6200 \cdot e^{2.64} \]
\[ A \approx 11518.68 \]
Therefore, the value of the quantity after 4 years, to the nearest hundredth, is approximately 11518.68.