Answer:
D. About 15.69 hours.
Step-by-step 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: [tex]y=e^{kx}[/tex], where,
e= mathematical constant,
k = Growth rate in decimal form.
Let us convert our given rate in decimal form.
[tex]7\%=\frac{7}{100}=0.07[/tex]
Upon substituting our given values we will get exponential function for bacteria growth as: [tex]y=2,000*e^{0.07x}[/tex], 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.
[tex]6,000=2,000*e^{0.07x}[/tex]
Let us divide both sides of our equation by 2,000.
[tex]\frac{6,000}{2,000}=\frac{2,000*e^{0.07x}}{2,000}[/tex]
[tex]3=e^{0.07x}[/tex]
Let us take natural log of both sides of our equation.
[tex]\text{ln}(3)=ln(e^{0.07x})[/tex]
[tex]\text{ln}(3)=0.07x[/tex]
[tex]1.0986122=0.07x[/tex]
[tex]x=\frac{1.0986122}{0.07}[/tex]
[tex]x=15.69446\approx 15.69[/tex]
Therefore, it will take about 15.69 hours to be 6,000 bacteria and option D is the correct choice.
