A quantity with an initial value of 3600 grows continuously at a rate of 2.5% per decade. What is the value of the quantity after 47 years, to the nearest hundredth?
To solve this problem, we can use the formula for continuous growth:
A = P * e^(rt)
Where A is the final amount, P is the initial amount, e is the base of the natural logarithm (approximately 2.71828), r is the growth rate, and t is the time in years.
In this problem, the initial amount is 3600, the growth rate is 2.5% per decade (or 0.025 per year), and the time is 47 years. Plugging these values into the formula, we get:
A = 3600 * e^(0.025 * 47)
Calculating the exponent, we find:
0.025 * 47 = 1.175
So the formula becomes:
A = 3600 * e^1.175
Using a calculator or a computer program to calculate e^1.175, we find:
e^1.175 ≈ 3.239
Multiplying this value by 3600, we get:
A ≈ 3600 * 3.239 ≈ 11660.4
Therefore, the value of the quantity after 47 years is approximately 11660.4 to the nearest hundredth.