Question

A bacteria population experiences continuous growth. If there were 2,000 bacteria present at the start of an experiment, how long did it take for there to be 6,000 bacteria if the growth rate was 7% per hour

about 0.06 hours

about 3.76 hours

about 6.82 hours

about 15.69 hours

Answer 1

D. About 15.69 hours.

Explanation:

Let x be the number of hours.

We have been given that there were 2,000 bacteria present at the start of an experiment and the growth rate was 7% per hour.

Since number of bacteria is growing exponentially, so we will use exponential growth function to solve our given problem.

The continuous exponential growth formula is in form:
`y=e^(kx)`, where,

e= mathematical constant,

k = Growth rate in decimal form.

Let us convert our given rate in decimal form.

`7\%=(7)/(100)=0.07`

Upon substituting our given values we will get exponential function for bacteria growth as:
`y=2,000*e^(0.07x)`, where, y represents number of bacteria after x hours.

Since we need to figure out the number of hours it will take for there to be 6,000 bacteria, so we will substitute y= 6,000 in our function.

`6,000=2,000*e^(0.07x)`

Let us divide both sides of our equation by 2,000.

`(6,000)/(2,000)=(2,000*e^(0.07x))/(2,000)`

`3=e^(0.07x)`

Let us take natural log of both sides of our equation.

`\text{ln}(3)=ln(e^(0.07x))`

`\text{ln}(3)=0.07x`

`1.0986122=0.07x`

`x=(1.0986122)/(0.07)`

`x=15.69446\approx 15.69`

Therefore, it will take about 15.69 hours to be 6,000 bacteria and option D is the correct choice.