Help with Continuous Growth

Question

Write the exponential function y=16(1.2)^t in the form y=ae^kt.

c) The continuous growth rate is ? % per year (round to the nearest 0.01%)

I got the answer 18 but the math homework marks it as wrong.

Answer 1

Is t the time in years? If it is, then the only problem I can see with your answer is that it is not rounded to he nearest 0.01%.

The continuous growth rate, rounded to the nearest 0.01%, is 18.23%.

Here's how the continuous growth rate is calculated.

Starting with y = 16(1.2)^t

To put this in the form y = ae^kt

We use the fact that (1.2)^t can be written as (e^ln(1.2))^t because e^ln(1.2) = 1.2

(e^ln(1.2))^t = e^{ln(1.2) × t} = e^kt where k = ln(1.2) = 0.1823

So 16(1.2)^t = 16e^0.1823t

k = ln(1.2) = 0.1823 is the continuous grown rate expressed as a decimal.

So, expressed as a percentage, the continuous grown rate is 18.23%

Answer 2

For this, k would be the rate which would be ln(1.2) = 0.182321556 = 18.2321556%. Rounding to the nearest 0.01% gives18.23%.