Question

In an insect colony, there are 230 insects after 7 days. If there were initially 100 insects, how long will it take the population to grow to 600 insects?

Answer 1

The population growth in this question is assumed to be exponential.

The exponential formula is given to be:

`y=a(b)^x`

where a is the initial amount and b is the growth rate of the population.

The question provides the following parameters:

`a=100`

We can find the growth rate by making the following substitution: If there were 230 insects after 7 days, we have:

`x=7,y=230`

Therefore,

`\begin{gathered} 230=100(b)^7 \\ b^7=2.3 \\ b=\sqrt[7]{2.3} \\ b=1.126 \end{gathered}`

Therefore, the exponential model for the problem is:

`y=100(1.126)^x`

To find the time it takes for the population to reach 600 insects, we can substitute for y = 600 into the model and solve for x:

`\begin{gathered} 600=100(1.126)^x \\ 1.126^x=(600)/(100) \\ 1.126^x=6 \end{gathered}`

Finding the natural logarithm of both sides:

`\ln 1.126^x=\ln 6`

Recall the law of logarithms:

`\ln x^a=a\ln x^{}`

Therefore,

`\begin{gathered} x\ln 1.126=\ln 6 \\ \therefore \\ x=(\ln 6)/(\ln 1.126) \\ x=15.098\approx15 \end{gathered}`

Therefore, it will take 15 days for the insect population to reach 600 insects.