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.
To find out when there will be 200,000 bacteria, we need to solve for x in the equation 𝑦 = 192(4^x-3) when y is equal to 200,000.
So, we set up the equation:
200,000 = 192(4^x-3)
Divide both sides by 192:
200,000/192 = 4^x-3
1,041.6667 = 4^x-3
Add 3 to both sides:
1,041.6667 + 3 = 4^x
4^x = 1,044.6667
Take the logarithm on both sides to solve for x:
log(4^x) = log(1,044.6667)
x * log(4) = log(1,044.6667)
x = log(1,044.6667) / log(4)
x ≈ 4.63
Therefore, there will be 200,000 bacteria approximately 4.63 hours after the culture is prepared. Rounded to the nearest whole hour, there will be 200,000 bacteria after 5 hours.