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:
, where,
e= mathematical constant,
k = Growth rate in decimal form.
Let us convert our given rate in decimal form.
![7\%=\frac{7}{100}=0.07](https://tex.z-dn.net/?f=7%5C%25%3D%5Cfrac%7B7%7D%7B100%7D%3D0.07)
Upon substituting our given values we will get exponential function for bacteria growth as:
, 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}](https://tex.z-dn.net/?f=6%2C000%3D2%2C000%2Ae%5E%7B0.07x%7D)
Let us divide both sides of our equation by 2,000.
![\frac{6,000}{2,000}=\frac{2,000*e^{0.07x}}{2,000}](https://tex.z-dn.net/?f=%5Cfrac%7B6%2C000%7D%7B2%2C000%7D%3D%5Cfrac%7B2%2C000%2Ae%5E%7B0.07x%7D%7D%7B2%2C000%7D)
Let us take natural log of both sides of our equation.
![\text{ln}(3)=ln(e^{0.07x})](https://tex.z-dn.net/?f=%5Ctext%7Bln%7D%283%29%3Dln%28e%5E%7B0.07x%7D%29)
![\text{ln}(3)=0.07x](https://tex.z-dn.net/?f=%5Ctext%7Bln%7D%283%29%3D0.07x)
Therefore, it will take about 15.69 hours to be 6,000 bacteria and option D is the correct choice.