The number of bacteria in a sample is increasing according to an exponential model. After two hours, the sample contained 200 bacteria. After six hours, the sample contained 800 bacteria.

Question

The number of bacteria in a sample is increasing according to an exponential model. After two hours, the sample contained 200 bacteria. After six hours, the sample contained 800 bacteria. Write an exponential growth model for the number of bacteria in the sample after x hours.

Answer 1

There's a couple of ways to do this. You can use a systems of equations.

Use the formula, P = a*b^x, where P is the population of bacteria and x is the time in hours.

We have two unknowns, a and b.

To solve, we'll plug in points of the form (x, P)

One point is (2, 200) (after 2 hours, the population is 200)

Therefore, an equation is 200 = a* b²

The second point is (6, 800).

Therefore, a second equation is 800 = a * b⁶

We'll write out the equations again:

800 = a * b⁶

200 = a* b²

We can divide these two equations.

On the left hand side, when we divide the equations, we get 800/200 = 4

On the right hand side, when we divide the equations, we get a*b⁶ / (a*b²) = b⁴

Therefore, 4 = b⁴

Solve for b:

b = ⁴√4 or 4^1/4

Next, solve for a, by plugging in b into the second equation

200 = a * (4^1/4)² = a * 4^1/2 = a * 2

Therefore, dividing both sides by 2, gives us that a = 100

Lastly, write out the equation with the known values of a and b. P = 100*(4^1/4)^x

Or, P = 100 * 4^x/4

Answer 2

Answer 3