Growth of a population of wolves

In summary, the function N(t)=(10N0)/(1+9e^(-t/4)) describes the growth of a population of wolves, where t is in years and N0 is the initial amount. To find the time it takes for the population to double, one can solve the equation N(t) = 2N0 for t. This is an exercise in algebra and not calculus, so any discrepancies between the calculated answer and the answer in the book may be due to a mistake or an error in the answer sheet. It is recommended to check the answer by plugging it back into the function.
  • #1
Emethyst
118
0

Homework Statement


A population of wolves grows according to the function N(t)=(10N0)/(1+9e^(-t/4)), where t is in years and N0 is the initial amount. How long from t=0 will the population take to double?



Homework Equations


Refer to below heading.



The Attempt at a Solution


Not sure if this is a completely calculus based question or not, but I threw it in here because it's listed in my calculus problems. Basically this one had stumped me, as I have no idea how to go about solving it. My first attempt was to try and find the initial population by using t=0, and then use that value to calculate the time it would take for it to double. This got me nowhere, for all I ended up with was N(t)=N0. I tried to use this by making N(t)=2N(t) and solving for t, but this only got close to the correct answer of 4ln10. Any help you guys can provide would be greatly appreciated on this question, thanks in advance :smile:
 
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  • #2
Emethyst said:

Homework Statement


A population of wolves grows according to the function N(t)=(10N0)/(1+9e^(-t/4)), where t is in years and N0 is the initial amount. How long from t=0 will the population take to double?



Homework Equations


Refer to below heading.



The Attempt at a Solution


Not sure if this is a completely calculus based question or not, but I threw it in here because it's listed in my calculus problems. Basically this one had stumped me, as I have no idea how to go about solving it. My first attempt was to try and find the initial population by using t=0, and then use that value to calculate the time it would take for it to double. This got me nowhere, for all I ended up with was N(t)=N0.
If you set t= 0 then, of course, you got N(0)= N0. That is the initial population. So you need to solve [itex]10N_0/(1+ 9e^{-t/4}= 2N_0[/itex] for t.

I tried to use this by making N(t)=2N(t) and solving for t, but this only got close to the correct answer of 4ln10. Any help you guys can provide would be greatly appreciated on this question, thanks in advance :smile:
Obviously, N(t)= 2N(t) has solution N(t)= 0! So that isn't what you mean. You really mean to solve N(a+ t)= 2N(a) for t, the time it take for the population at t= a to double.
But your first idea works: divide both sides of [itex]10N_0/(1+ 9e^{-t/4}= 2N_0[/itex] by N0 to get [itex]10/(1+ 9e^{-t/4}= 2[/itex] and then [itex]10= 2+ 18e^{-t/4}[/itex]. But we can't say what you did wrong if you don't tell us what you did!
 
  • #3
Could you fix the latex images HallsofIvy? It says there is an error in them :tongue:
 
  • #5
I solved N(t)=2N0 and came up with the answer of -4ln(4/9), which is the answer I came up with before (I just had my variables mixed up), so I am still guessing I am still missing something here.
 
  • #6
This is basically an exercise in algebra, not calculus, so if you're not getting the answer that's posted, then you've made a mistake or the answer in the book is wrong.

You can check both answers by plugging them in your function. A correct value of t should produce 2N0.
 
  • #7
My answer is correct, I just plugged it in and got 2N0, so the answer sheet must be wrong for that question. Thanks for the help Mark and HallsofIvy, even if it was a more algebra based question :tongue:
 

1. What factors contribute to the growth of a population of wolves?

The growth of a population of wolves is influenced by several factors including availability of prey, suitable habitat, competition with other species, and human interference. These factors can impact the wolves' ability to find food, reproduce, and survive in their environment.

2. How do wolves reproduce and how often?

Wolves typically reproduce once a year with a gestation period of around 63 days. Mating usually occurs in late winter or early spring, and the female gives birth to a litter of 4-6 pups. The pups stay with their parents for up to 2 years before dispersing to form their own packs.

3. What is the average size of a wolf pack?

The average size of a wolf pack varies depending on the region and availability of resources, but it typically ranges from 5 to 10 individuals. However, packs can also consist of just a breeding pair and their offspring or can grow to include up to 30 individuals in some cases.

4. How do human activities affect the growth of wolf populations?

Human activities such as habitat destruction, hunting, and poisoning have significantly impacted wolf populations in the past. However, conservation efforts and protective measures have helped to increase the populations of wolves in some areas. Human-wolf conflicts, such as livestock depredation, can also disrupt wolf populations if not managed properly.

5. What are the consequences of an unsustainable growth of wolf populations?

An unsustainable growth of wolf populations can lead to overpopulation, which can have negative impacts on other species and ecosystems. It can also result in conflicts with humans, as wolves may start preying on livestock or pets in search of food. In these cases, population control measures may be necessary to maintain a balance between wolves and their environment.

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