Question

Researchers recorded that a certain bacteria population declined from 750,000 to 250 in 48 hours after the administration of medication. At this rate of decay, how many bacteria will there be in 8 hours?

Answer 1

There will be 66 bacteria in 8 hours.

Explanation:

The number of bacteria after t hours is given by the following formula.

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

In which P(0) is the initual number of bacteria and r is the decay rate.

Researchers recorded that a certain bacteria population declined from 750,000 to 250 in 48 hours after the administration of medication.

This means that
`P(0) = 750000, P(48) = 250`

We use this to find r. So

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

`250 = 750000(1-r)^(48)`

`(1-r)^(48) = (250)/(750000)`

`\sqrt[48]{(1-r)^(48)} = \sqrt[48]{(250)/(750000)}`

`1-r = 0.84637`

So

`P(t) = 750000(0.84637)^(t)`

How many bacteria will there be in 8 hours?

8 hours from now, in this context, is 8 + 48 = 56 hours. So this is P(56).

`P(56) = 750000(0.84637)^(56) = 65.83`

Rounding to the nearest number

There will be 66 bacteria in 8 hours.

Answer 2

197,488

Step-by-step explanation:

This problem requires two main steps. First, we must find the unknown rate, k. Then, we use that value of k to help us find the unknown number of bacteria.

Identify the variables in the formula.

AA0ktA=250=750,000=?=48hours=A0ekt

Substitute the values in the formula.

250=750,000ek⋅48

Solve for k. Divide each side by 750,000.

13,000=e48k

Take the natural log of each side.

ln13,000=lne48k

Use the power property.

ln13,000=48klne

Simplify.

ln13,000=48k

Divide each side by 48.

ln13,00048=k

Approximate the answer.

k≈−0.167

We use this rate of growth to predict the number of bacteria there will be in 8 hours.

AA0ktA=?=750,000=ln13,00048=8hours=A0ekt

Substitute in the values.

A=750,000eln13,00048⋅8

Evaluate.

A≈197,488.16

At this rate of decay, researchers can expect 197,488 bacteria.