The number of bacteria in a culture is increasing according to the law of exponential growth. After 3 hours, there are 100 bacteria, and after 6 hours, there are 800 bacteria.

Question

The number of bacteria in a culture is increasing according to the law of exponential growth. After 3 hours, there are 100 bacteria, and after 6 hours, there are 800 bacteria. How many bacteria will there be after 7 hours?

Answer 1

The generic formula for exponential growth is y = y₀*b^x, where y₀ and b are the two unknowns. If we plug in our two unknown states, we get:

100 = y₀*b³

800 = y₀b⁶.

Dividing the bottom equation by the top gives us:

800/100 = (y₀b⁶)/(y₀b³)

8 = b³

b = 2

In other words, our colony population is multiplying by 2 every hour. After 7 hours, then, we simply have 1,600 bacteria.

If we need to find the whole equation, we can now just plug in b to either data point above. Choosing the first one gives 100 = y₀*2³ = y₀*8, or y₀ = 100/8. So in general, y = 12.5*2^x. Hope this helps!

Answer 2

Law of exponential growth means the the number of bacteria, y, after a time period t is given by the exponential model y=ab^t, where b>1. (exponential decay is when 0<b<1).

After 3 hours there's 100 bacteria translates to t=3 and y=100:

y=ab^t

100=ab^3

a=100/b^3 <-plug this into the exponential model y=ab^t

y=ab^t becomes:

y=100/b^3 (b^t)

So far the exponential model is of the form y=100/b^3 (b^t).

Now we use the second condition:

Then after 6 hours, there's 800 bacteria, so t=6, and y=800. Plugging that in allows us to solve for b:

y=100/b^3 (b^t)

800=100/b^3 (b^6)

800=100b^3

8=b^3

b^3=8

b=2 <--- plug this into the exponential model y=100/b^3 (b^t)

y=100/2^3(2^t)

y=100/8 (2^t)

Now this allows you to predict the population of bacteria at any point in time t. So plug in t=7 into the exponential model to get: y=100/8(2^7)=1600 bacteria.