If I start off with 1 bacteria, and it triples every hour, how many do I have after 24 hours?

Question

I'm in 9th grade, and I'm having some difficulties completing a math assignment in my Algebra 1 class. My question is about exponential growth.

If I start off with 1 bacteria, and it triples every hour, how many will I have after 24 hours? If you answer this, please walk me through the steps. It would be greatly appreciated!

One more question. What would the formula for this situation be? ($y=a(1+r)^x$)

Answer 1

Consider what is actually happening in the question.

You begin with 1 bacteria, and after 1 hour it has tripled so you now have $1$ x $3=3$.

Next hour, it triples again so you now have $1$ x $3$ x $3=9$

You can see the pattern shows that the number of bacteria is multiplying every hour by a factor of 3. An exponent denotes how many times we are multiplying a number by itself, for example: $3^4$ means we are multiplying the number 3 a total of 4 times ($3$ x $3$ x $3$ x $3$).

Therefore the question is requiring us to triple the number of bacteria every hour for 24 hours, which means we are multiplying by 3 a total of 24 times. This gives us:

$n$ x $3^{24}$ where n is the number of bacteria you begin with.

Since you begin with 1 bacteria, the solution is $1$ x $3^{24}=3^{24}$

Answer 2

Let $a$ be the initial amount of bacteria you have. Then each hour you are going to multiply by $3$. So it is going to look like $a(3)$ for $1$ hour, $a(3)(3)$ for $2$ hours, $a(3)(3)(3)$ for $3$ hours and so on. In order to simplify this you can say you have $a(3)^x$ where $x$ is the amount of hours that pass. The formula for this will be $y=a(3)^x$.

The formula you gave is the exponential growth function where $a$ is the initial amount, $x$ is the amount of hours (or whatever time interval you are using) passes, and $r$ is the growth rate that is given in a percentage/decimal.

In this case, $r$ would be $200$% or $2.00$. Then you would have $y=a(1+2.00)^x=a(3)^x$.

Hope this helps!

Answer 3

Assume the amount of bacteria is a function of the form $y(x)=A e^{rx}$, with $r,A$ unknown, and $x$ being time in hours. Since we start off with 1 bacteria, this means $y(0)=Ae^0=A=1.$ The statement about it tripling every hour means the following. Fix a time $x$. Then $y(x+1)=3y(x)$. In other words, with $A=1$, we have $e^{r(x+1)}=3e^{rx}.$ Use the fact that $e^{r(x+1)}=e^{rx}e^r.$ Then the $e^{rx}$ cancel on both sides and we get $e^r=3$.

Hence, $r=\ln(3)$. Therefore, the original function is $y(x)=e^{\ln(3)x}$, (which actually reduces to $(e^{\ln(3)})^x=3^x.$) Can you finish the rest?