Section 3.5 Continuous Growth

b) We substitute the \(a\) and \(k\) found in part a) into the equation \(P(t)=ae^{kt}\text{.}\)

a) Recall the compound interest formula \(A(t) = P \left(1 + \frac{r}{n}\right)^{nt}\text{,}\) where \(P\) is the principal amount invested, \(r\) is the nominal interest rate, \(n\) is the number of times interest is compounded, and \(t\) is the number of years interest is applied.

b) Recall the continuous growth formula \(A(t)=Pe^{rt}\text{,}\) where \(P\) is the principal amount invested, \(r\) is the nominal interest rate, and \(t\) is the number of years interest is applied.

You want to evaluate these equations at \(t=9\) once you’ve filled in the appropriate values.

a) i) Plug in \(t=0\text{.}\)

ii) Try to find the growth factor or consider what the growth rate is. Recall that if the growth factor is larger than 1, the quantity is increasing. If the growth factor is less than 1, the quantity is decreasing.

Recall that \(b\) is the growth factor in the equation \(y=a(b)^t\text{,}\) so you want to figure out what \(b\) is to determine the growth factor.

In terms of growth rates, positive growth rates indicate increasing, which negative growth rates indicate decreasing.

iii) The effective annual rate can be found by subtracting 1 from the growth factor. Remember to convert this into a percentage before entering it into the answer box.

iv) Recall that \(A(t)=Pe^{rt}\) gives the continuous growth formula.

b) Use a similar process to part a) to answer each of these questions.

a)

i) Recall the formula for annual growth: \(A(t) = P (1+r)^t\text{,}\) where \(P\) is the principal amount invested and \(r\) is the nominal interest rate.

ii) Recall the formula for continuous growth: \(A(t) = P (e)^{rt}\text{.}\)

For both parts i) and ii), we want to evaluate the equations above at \(t=15\text{.}\)

b) Compare your answers to parts i) and ii).

a) We are looking for a formula of the form \(P(t) = P (e)^{rt}\text{,}\) where \(P\) is the initial population and \(r\) is the continuous growth rate.

b) Recall the effective growth rate is the growth factor minus 1. You’ll need to find the growth factor in this case (which is not just \(e\)). Remember to convert this to a percentage.

c) Think about compounding interest. Why is the effective growth rate larger for continuous compound when compared with compounding annually?

a) Usually we think about a growth rate as how much a population grows in a year. Here, the question wants us to consider the growth rate as how much the population grows in an hour. So, we can use the same formula for annual growth, where \(t\) is instead in hours instead of years. That is, we use the formula \(B(t) = P (1 + r)^t\text{,}\) where \(B(t)\) is the population of the bacteria after \(t\) hours instead of \(t\) years. \(P\) is the initial population and \(r\) is the growth rate, as usual.

b) As in part a), since this is growth rate for an hour, we use the formula for continuous growth, where \(t\) is instead in hours instead of years. That is, we use the formula \(B(t) = P e^{rt}\text{,}\) where \(B(t)\) is the population of the bacteria after \(t\) hours instead of \(t\) years. Again, \(P\) is the initial population and \(r\) is the growth rate.

For parts a) and b), we want to evaluate the function at \(t=12\) for the second part.

c) Find the growth factors and compare.

a) We are looking for a formula of the form \(I(t) = P e^{rt}\text{,}\) since growth is continuous. \(P\) is the initial number of subscribers in 2000 and \(r\) is the continuous growth rate.

Note that since the output is in millions of people, you don’t need to indicate millions when writing your formula.

b) The effective annual growth rate is how much the number of subscribers changes given a one year increase in time. Recall, for continuous growth, the growth factor is \(e^r\) and the effective growth rate is given by the growth factor minus 1. Remember to convert to a percentage.

b) Use the same process as in part a).

c) Look at the trend in part a) and b).