Question

In a parenting magazine, Mrs. Smith read an advertisement about all the different ways that bleach can be used to kill bacteria. The advertisement claimed that some forms of bacteria triple in amount after each hour. Consider the situation where one bacterium is found on Mrs. Smith's countertop at home.

Answer 1

Question:

(a) How many bacteria would be found in 24 hours

(b) How many bacteria would be found in 2 days

(c) How long for 1000 bacteria to be found

Answer:

(a) 282429536481 bacteria

(b)
`7.97*10^{22` bacteria

(c) 6 days

Explanation:

The question illustrates an exponential function

`y = ab^x`

Where

`a = Initial\ Value = 1`

`b=Rate = 3` --- i.e. triples

`x = number\ of\ hours`

Solving (a): Bacteria in 24 hours

In this case:

`x = 24`

Substitute
`x = 24`,
`b = 3` and
`a = 1`

So:

`y = ab^x`

`y = 1 * 3^{24`

`y = 1 * 282429536481`

`y = 282429536481` bacteria

Solving (b): Bacteria in 2 days

`2\ days =48\ hours`

So:

`x = 48`

Substitute
`x = 48`,
`b = 3` and
`a = 1`

So:

`y = ab^x`

`y = 1 * 3^{48`

`y =7.97*10^{22` bacteria

Solving (c): How long for 1000 bacteria

In this case:

`y = 1000`

Substitute
`y = 1000`,
`b = 3` and
`a = 1`

So:

`y = ab^x`

`1000 = 1 *3^x`

`1000 = 3^x`

Take Log of both sides

`Log1000 = Log3^x`

This gives:

`Log1000 = xLog3`

Make x the subject

`x = (Log1000)/(Log3)`

`x \approx 6`

Hence: It takes 6 days