Question

Researchers recorded that a certain bacteria population declined from 200,000 to 900 in 18 hours. At this rate of decay,

how many bacteria was there at 10 hours? Round to the nearest whole number,

Answer 1

9,942 bacteria were there at 10 hours.

Explanation:

Equation for population decay:

The equation for population decay, after t hours, is given by:

`P(t) = P(0)(1-r)^t`

In which P(0) is the initial population and r is the decay rate, as a decimal.

Researchers recorded that a certain bacteria population declined from 200,000 to 900 in 18 hours.

This means that
`P(0) = 200000` and that when
`t = 18, P(t) = 900`. So we use this to find r.

`P(t) = P(0)(1-r)^t`

`900 = 200000(1-r)^(18)`

`(1-r)^(18) = (900)/(200000)`

`\sqrt[18]{(1-r)^(18)} = \sqrt[18]{(900)/(200000)}`

`1 - r = ((900)/(200000))^{(1)/(18)}`

`1 - r = 0.7407`

So

`P(t) = P(0)(1-r)^t`

`P(t) = 200000(0.7407)^t`

At this rate of decay, how many bacteria was there at 10 hours?

This is P(10). So

`P(10) = 200000(0.7407)^(10) = 9941.5`

Rounding to the nearest whole number:

9,942 bacteria were there at 10 hours.