In the formula N = le^kt, N is the number of items in terms of an initial population l at a given...

Question:

In the formula {eq}N = le^kt {/eq}, {eq}N {/eq} is the number of items in terms of an initial population {eq}l {/eq} at a given time {eq}t {/eq} and {eq}k {/eq} is a growth constant equal to the percent of growth per unit time. How will it take for the population of a certain country to triple if its annual growth rate is 2.4%?

Exponential Growth

If a quantity grows exponentially, then it grows at a constant percentage rate. We can also define this growth in the amount of time that it would take for the quantity to double, triple, or reach another multiple of its initial value. This is because regardless of the initial value, an exponential function will have the same doubling time corresponding to a specific percentage growth rate.

{eq}f(t) = p_0 e^{kt} {/eq}

Answer and Explanation:

We've been told two pieces of information. First, that the annual percent growth rate is 2.4%. Second, we've been told the model to use to represent this population. If we assume that the annual percent growth rate is the continuous annual percent growth rate, then we can use this 2.4% to represent the growth constant, k. This makes our model have the following form.

{eq}N = I e^{0.024t} {/eq}


If the initial population were to triple, then the population would have a value of 3I. We can thus set this function equal to 3I and solve for the value of time it would take to reach this value. This would be the tripling time.

{eq}3I = I e^{0.024t}\\ 3 = e^{0.024t}\\ \ln 3 = \ln e^{0.024t }\\ 0.024 t = \ln 3\\ t = \frac{1}{0.024} \ln 3 \approx 45.7755 {/eq}


Therefore, the population will triple after approximately 45.78 years.


Learn more about this topic:

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Exponential Growth: Definition & Examples

from High School Algebra I: Help and Review

Chapter 6 / Lesson 10
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