Question

A population of 50 triples in size every 5 years. How long will it take the population to reach 6,250,000 people?Show your work please!

Answer 1

We know that the population triples in size every 5 years and that we start with 50, that means that:

`\begin{gathered} p(0)=50 \\ p(5)=3p(0)=3\cdot50 \\ p(10)=3p(5)=3\cdot3\cdot50=3^2\cdot50 \\ p(15)=3p(10)=3\cdot3^2\cdot50=3^3\cdot50 \end{gathered}`

from this pattern we notice that that the population in general will be of the form:

`p(t)=3^{(t)/(5)}*50=(50)3^{(t)/(5)}`

Hence the population is given by:

`p(t)=(50)3^{(t)/(5)}`

Now that we have this expression we can find how long it will take to reach 6,250,000. To do this we equate the function to the population we want and solve for t:

`\begin{gathered} (50)3^{(t)/(5)}=6250000 \\ 3^{(t)/(5)}=(6250000)/(50) \\ 3^{(t)/(5)}=125000 \\ \log _3(3^{(t)/(5)})=\log _3(125000) \\ (t)/(5)=\log _3(125000) \\ t=5\log _3(125000) \end{gathered}`

to find this number with a calculator we can use the change o base formula:

`\log _bx=(\ln x)/(\ln b)`

Then in our case, we have:

`t=(5\ln 125000)/(\ln 3)`

plugging this expression into a calculator we get:

`t=53.413`

Therefore it will take approximately 53.413 years to reach that population.