A bacterial culture quadruples in size every hour. You start to observe
the number of bacteria 3 hours after the culture is prepared. The
amount of “y” of bacteria “x” hours after the culture is prepared is
represented by 𝑦 = 192(4
𝑥−3
). When will there be 200,000 bacteria?
Use your calculator and include the steps you took to find your
answer. Round your answer to the nearest whole hour. Without using the log function
Given that the equation representing the amount of bacteria present after x hours is y = 192(4^x-3), we need to find the value of x when y = 200,000.
Setting y = 200,000, we have:
200,000 = 192(4^x-3)
Divide both sides by 192:
200,000 / 192 = 4^x-3
1041.67 = 4^x-3
Add 3 to both sides:
1044.67 = 4^x
Now we need to find the value of x. Since the growth rate is quadrupling every hour, we can express 1044.67 as a power of 4. By trial and error, we can find that:
4^5 = 1024
4^6 = 4096
So x is between 5 and 6. To refine our estimate, we can try a value between 5 and 6:
4^5.5 = 2048
Therefore, x is approximately 5.5 hours. Rounding to the nearest whole hour, we can conclude that there will be 200,000 bacteria after 6 hours.