Question

A population of bacteria is initially 6000. After three hours the population is 3000. If this rate of decay continues, find the exponential function that represents the size of the bacteria population after t hours. Write your answer in the form f(t)=a(b)t.

Answer 1

`f(t) = 6000 ((1)/(2))^t^/^3`

Explanation:

Let's take the equation:

`y = y_0 e^-^k^t`

Given the initial population of bacteria = 6000,

`y = 6000e^-^k^t`

Population of bacteria after 3 hours is 3000.

y(3) = 3000

Thus,

`6000e^-^k^3 = 3000`

`e^-^k^3 = (3000)/(6000)`

`e^-^k^3 = (1)/(2)`

`(e^-^k)^3 = (1)/(2)`

`e^-^k= ((1)/(2))^1^/^3`

Let's substitute
`((1)/(2))^1^/^3` for
`e^-^k` in
`y = 6000e^-^k^t`

Therefore, we have:

`6000e^-^k^t`

`= 6000 [((1)/(2)) ^1^/^3]^t`

`= 6000 [(1)/(2)] ^t^/^3`