The bacteria on Bailey’s petri dish are doubling in area every $20$ minutes! The area of the dish covered by bacteria was initially $1$ cm^{$^2$}, then $2$ cm^{$^2$}, then $4$ cm^{$^2$}, then $8$ cm^{$^2$}, etc. The science teacher asks Bailey’s team to predict the area that will be covered by bacteria after two hours.

1. Each team member writes a different expression for his or her prediction. Bailey writes $1(2)^6$. Carmen writes $1(2^{1/20})^{120}$. Demetri writes $1(2^3)^2$. Explain what the numbers in each expression mean in terms of the situation. What area will be covered by bacteria after two hours?

   Answer (a):

   $1$ is the starting number of bacteria. Bailey’s expression shows that the bacteria multiply by $2$ (every $20$ minutes), and because there are six $20$-minute periods in two hours, the multiplier needs to be applied $6$ times. Carmen’s expression shows that the number of bacteria will multiply by $2^{1/20}$ per minute, and there are $120$ minutes in two hours. Demetri’s expression shows that the number of bacteria will multiply by $2^3$ every hour during the two-hour period. There will be $64{,}000$ bacteria after two hours.

2. Now Bailey’s team has to determine the area that would have been covered by bacteria $10$ minutes before they started the experiment. Write two different expressions that represent this situation and then determine the answer.

   Partial Answer (b):

   $≈ .707$ sq. cm
   What are two expression that give this answer?