Growth Rate. A precise definition.

Question

Recently I came across a problem (see statement below) growth rate . During my attempts to solve the exercise I concluded that I do not know the meaning of need backup growth rate when this rate is noconstant and varies finds a discrete parameter.

Problem ( growth rate ). The current circulation a certain journal is 5000 copies per week. Knowing that owner of Journal projects a growth rate of $6+8\cdot t^{\frac{3}{5}}$ copies to week, hence the $t$ weeks, over the next five years, the newspaper circulation in about 150 weeks is how many copies ?

Update. The answer suggested by the source of this problem is 20.050 copies

It seems to me that understanding the precise meaning of growth rate is determining how to suitable the formula to solve the exercise.

I did an internet search and found multiple meanings that lead to various formulas : population growth , birth rate, death rate , net profit rate , etc. ...

Observation. I have no problem as algebraic manipulations.

My first question is: What is the concept of growth rate that is suitable for use in the exercise below ? How to apply it to solve the problem? This concept is just a convention ?

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Second question: What are the different concepts of rate of growth that exist and how to use them ? Is there any reference?

Answer 1

You are correct, it is not a well-defined term. I can see the following ways to read the problem:

1. the circulation in week $t$ is $6+8t^{\frac 35}$ more copies than in week $t-1$. I would use this one, so you just need to sum $6+8t^{\frac 35}$ from $1$ to $150$ and add $5000$ to get your answer.
2. the circulation in week $t$ is $1+(6+8t^{\frac 35})$ times that in week $t-1$. This doesn't make sense with this value, but if you were given a $10\%$ growth rate it would
3. the circulation in week $t$ is $6+8t^{\frac 35}$ total copies. This doesn't match your $5000$ value for a reasonable value of $t$. I wouldn't call this a growth rate, but I have seen it done.

Answer 2

Growth rate means derivative in every example I have ever seen. When used in a discrete situation, as here, it is assumed that we are using a continuous model of a discrete situation. This is very common in applied math and physics, like thermodynamics.