Exponential Models with Differential Equations

Question

A colony of bacteria grows under ideal conditions so that the population increases exponentially with time. After 2 hours there are 500 bacteria. After 6 hours there are 10,000 bacteria. How many bacteria were present initially? Show all necessary steps.

Answer 1

500 = a₀b²

10000 = a₀b⁶

20 = b⁴, divide bottom equaton by to equation

⁴√20 = b

Now substitute b into either origianl equation and solve for a₀

Answer 2

An exponential equation is in the form y=a(b)^x With the data points given, you can make 2 equations. I prefer to solve this by substitution method.

500=a(b)².

10000=a(b)⁶

Using the first equation, I will solve for a.

a=500/b²

Then plug that into the second equation.

10000=(500/b²)(b⁶)

Simplify the b terms. The exponent subtract for division.

10000=500b⁴

Divide both sides by 500.

20=b⁴

^{Take the 4th root of both sides}

b=⁴√20

Then, you can solve for a, the initial value. From our previous equation:

a=500/b²

a=500/(⁴√20)²

a=118.80 is the initial number of bacteria