Question

The number of bacteria in a culture is increasing according to the law of exponential growth. The initial population is 240 bacteria, and the population after 9 hours is double the population after 1 hour. How many bacteria will there be after 4 hours? (Round your answer to the nearest whole number.)

Answer 1

339

Explanation:

Exponential Function

General form of an exponential function:
`y=ab^x`

where:

• a is the initial value (y-intercept)
• b is the base (growth/decay factor) in decimal form
• x is the independent variable
• y is the dependent variable

If b > 1 then it is an increasing function

If 0 < b < 1 then it is a decreasing function

Given information:

• a = 240 (initial population of bacteria)
• x = time (in hours)
• y = population of bacteria

Therefore:
`y=240b^x`

To find an expression for the population after 1 hour, substitute x = 1 into the found equation:

`\implies y=240b^1`

`\implies y=240b`

We are told that the population after 9 hours is double the population after 1 hour. Therefore, make y equal to twice the found expression for the population after 1 hour, let x = 9, then solve for b:

`\implies 2(240b)=240b^9`

`\implies 480b=240b^9`

`\implies 480=240b^8`

`\implies 2=b^8`

`\implies b=\sqrt[8]{2}`

`\implies b=2^{(1)/(8)}`

Therefore, the final exponential equation modelling the given scenario is:

`\implies y=240(2^{(1)/(8)})^x`

`\implies y=240(2)^{(1)/(8)x}`

To find how many bacteria there will be after 4 hours, substitute x = 4 into the found equation:

`\implies y=240(2)^{(1)/(8)(4)}`

`\implies y=240(2)^{(1)/(2)}`

`\implies y=339 \:\: \sf (nearest\:whole\:number)`

Therefore, there will be 339 bacteria (rounded to the nearest whole number) after 4 hours.