Exponential Growth

Another application of exponential function is Exponential Growth. If half-life applies to radioactive elements, this topic is usually about bacteria. Think of one bacterium, after a certain period of time it doubles itself, after another period of time the two bacteria double themselves. Now there are already 4 of them. After another period of time there are 8 of them and so on. But the normal thing is, there is not just one bacterium. There is usually a colony of them. Using exponential functions we can mathematically determine the number of bacteria in the colony after a certain period of time.

Formula:

Given an initial number of bacteria in a colony $A_i$ and multiply itself $n$ times after time $t$. The number of bacteria (A_n) in the colony after time $T$ is given by the following formula:

$A_n=A_i(n)^{\frac{T}{t}}$

Sample Problem 1:

A fictional bacterium called Chenis madakis doubles itself every 15 minutes; a colony of sample containing 100 healthy bacteria is under observation, how many bacteria are there at the end of one hour?

[toggle title=”Solution”]

Using the formula above,

$A_n=A_i(n)^{\frac{T}{t}}$ $A_n=(100)(2)^{\frac{60}{15}}$ $A_n=(100)(2)^4$

$A_n=1600$ [/toggle]

Sample Problem 2:

A certain type of bacteria multiplies itself 3 times every 4 hours. Find the size of the sample of the colony after 1 day if initially there are 1000 bacteria.

[toggle title=”Solution”]

$A_n=A_i(n)^{\frac{T}{t}}$ $A_n=(1000)(3)^{\frac{24}{4}}$ $A_n=(1000)(3)^6$

$A_n=729,000$ [/toggle]

Sample Problem 3:

Celia kakaskakas doubles itself every 5 minutes. How long will it take for it to make itself 4 times as many as its original number?

[toggle title=”Solution”]

$A_n=4(A_i) \to \displaystyle\frac{A_n}{A_i}=4$   eqn.1

$A_n=A_i(n)^{\frac{T}{t}}$ $\displaystyle\frac{A_n}{A_i}=n^{\frac{T}{t}}$

Substitute from equation 1:

$4= 2^{\frac{T}{5}}$ $2^2=2^{\frac{T}{5}}$ $2=\displaystyle\frac{T}{5}$

$t=10$ minutes [/toggle]

Dan

Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.

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