Question
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?
Answers
GPT 3.5
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.
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.