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Hi!

First of all, it is important to understand that you have an exponential function. Therefore, we will be working with this equation :

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where f(x) is the number of ants and x the number of days since the start of the study.

We know that the population doubles in size every 4 days. This information gives us our parameters b and c, which are respectively 1/4 and 2. Indeed, our multiplier is 2 since the population doubles

$$ c=2$$

and the event occurs every 4 days, but the variable x is in days. There is 1 event every 4 days.

$$ b = \frac{1}{4}$$

Which gives us so far this equation :

$$ f(x) =a(2)^{\frac{1}{4}x}$$

$$ f(x) =a(2)^{0,25x}$$

We want to find how many ants were in the colony at the start of the study, therefore the value of our parameter \(a\), the initial value. We know that after 5 days, there are 6400 ants, which gives us the point (5, 6400). To find the value of \(a\), we can insert the coordinates of this point into our equation, like this :

$$ 6400 =a(2)^{\frac{1}{4} \times 5}$$

We will then be able to find the value of \(a\), our initial number of ants.

Here's a page that shows similar examples : Problem Solving with the Exponential Function | Secondaire | Alloprof

I hope I was able to help you! Feel free to ask us more questions if you have any! :)