How to know when to use continuously compounding formula?

[BigZero1]

I was under the impression that population growth always uses continuous growth such as: A=Pe^rt. However, I have the following question in my text about population that doesn't use that.

Question:
The fox population in a certain region has an annual growth rate of 9% per year. In the year 2012, there were 23,900 fox counted in the area. What is the fox population predicted to be in the year 2020?

This question is very simple to set up and answer as long as you know what to use. So I'm wondering how to know what to use, will a continuous growth question always state that it's continuous?

 

[Muddyakka]

Why is this not a continuos growth question?
A = Px^kt ,
This still applies to this question where t is in years. If you were to input 0.5 years you would still get a value for population.

However not all population will grow according to this. There exists formulas for modified growth and decay, but they will specify and provide you with relevant equations/expression.

To answer your question, most population questions will be in the form of A = Px^kt. If not, they will provide you with relevant information to be able to deduce another expression.

 

[JeffM]

I was under the impression that population growth always uses continuous growth such as: A=Pe^rt. However, I have the following question in my text about population that doesn't use that.

Question:
The fox population in a certain region has an annual growth rate of 9% per year. In the year 2012, there were 23,900 fox counted in the area. What is the fox population predicted to be in the year 2020?

This question is very simple to set up and answer as long as you know what to use. So I'm wondering how to know what to use, will a continuous growth question always state that it's continuous?

Your equation is about CONTINUOUS growth. You were asked a question about annual growth. That is the danger of relying on formulas.

 

[Cubist]

@BigZero1 I don't want to confuse you, so please ignore this post if you don't understand it within a minute or two (I don't know what level of math you are)!

The two formulas are pretty similar in nature, as @Muddyakka was saying. In fact it's possible to change \(A=Px^{kt}\) to the \(A=Pe^{rt}\) form if you choose \(r=k\ln(x)\). Perhaps you could show/ prove this?

 

[JeffM]

@Cubist and @Muddyakka

You are of course both correct. I did not mention this because (a) I just hate conveying the notion that math is memorizing a bunch of formulas and algorithms, and (b) I greatly doubted that someone who cannot distinguish continuous compunding from annual compounding would grasp the derivation of k.

If, however, my response was viewed as a criticism of Muddyakka, that was unintentional, and I apologize.

P.S. Of course I have memorized formulas and algorithms if I need to use them frequently and can derive them so I understand when they apply. But mathematical formulas are the result of math rather than math itself. Getting off my soapbox now.

 

[Deleted member 4993]

I was under the impression that population growth always uses continuous growth such as: A=Pe^rt. However, I have the following question in my text about population that doesn't use that.

Question:
The fox population in a certain region has an annual growth rate of 9% per year. In the year 2012, there were 23,900 fox counted in the area. What is the fox population predicted to be in the year 2020?

This question is very simple to set up and answer as long as you know what to use. So I'm wondering how to know what to use, will a continuous growth question always state that it's continuous?

.....how to know what to use.....

To answer the question above, the suggestion is to READ the question carefully - it may be explicitly stated like below:

...has an ANNUAL growth rate of....

If it gives you annual growth rate - it is discrete and NOT continuous by definition (since the growth will be calculated every year and NOT every Planck second).)

 

[BigZero1]

Thanks all, I think it came down to not reading the question good enough and having the idea in my head that all population growth was automatically continuous. Because of that assumption I just jumped straight to that formula. Will be more careful, thanks again.

 

[Cubist]

@Cubist and @Muddyakka

You are of course both correct. I did not mention this because (a) I just hate conveying the notion that math is memorizing a bunch of formulas and algorithms, and (b) I greatly doubted that someone who cannot distinguish continuous compunding from annual compounding would grasp the derivation of k.

If, however, my response was viewed as a criticism of Muddyakka, that was unintentional, and I apologize.

P.S. Of course I have memorized formulas and algorithms if I need to use them frequently and can derive them so I understand when they apply. But mathematical formulas are the result of math rather than math itself. Getting off my soapbox now.

No, I didn't view your post as any kind of criticism at all. And I thoroughly agree that memorising formulas isn't math. I've got a memory like a sieve, therefore I memorise a formula as a LAST RESORT. I'm an avid puzzle solver/ thinker which has really helped my journey in mathematics without relying on lots of formulas! I just thought that OP might be interested if I showed the link between the formulas.

...If it gives you annual growth rate - it is discrete and NOT continuous by definition (since the growth will be calculated every year and NOT every Planck second).)

I look forward to seeing the (approx) 0.247 of a fox that appeared on the stroke of midnight on the first day of 2020, since all foxes must give birth on that day/ time . Maybe the hounds got hold of it ?

 

[Muddyakka]

If, however, my response was viewed as a criticism of Muddyakka, that was unintentional, and I apologize.

No need to apologise. I can see where you are coming from hahaha. I did A = Px^rt, as I thought it is the result of a pretty intuitive deduction in forming an equation to worded problems. You have 23900 and you know it increases by a factor of 1.09 every year. This means A₁ = 23900 x (1.09), A₂ = 23900 x (1.090)², and so finding the formula in the form of A = Px^rt should result after some thinking.

However, I must agree that formulas should only be used and memorised once you can do the math without them and only as a method to save time in exams. This way solving problems isnt a matter of 'which formula should i use?' but rather 'have i seen something like this before?'.

Ultimately the ability to recognise what method/formula to use comes down to experience and fundamental math knowledge. @BigZero1