Bacteria colonies can increase by [tex]$73 \%$[/tex] every 2 days. If you start with 150 bacteria microorganisms, how large would the colony be after 8 days?
Sure! Let's break down the problem into detailed steps to find how large the bacteria colony would be after 8 days.
1. Identify Initial Amount (I): - The bacteria colony starts with 150 microorganisms. [tex]\[ I = 150 \][/tex]
2. Determine Growth Rate (r): - The growth rate is given as 73%, which is [tex]\( 0.73 \)[/tex] when expressed as a decimal. [tex]\[ r = 0.73 \][/tex]
3. Total Time (t): - The colony is observed over a period of 8 days. [tex]\[ t = 8 \text{ days} \][/tex]
4. Periods: - The growth rate is given per every 2 days. [tex]\[ \text{One period} = 2 \text{ days} \][/tex]
5. Number of Periods (n): - To determine how many periods fit into the total time, divide the total time by the length of each period. [tex]\[ n = \frac{t}{\text{Period}} = \frac{8 \text{ days}}{2 \text{ days}} = 4 \][/tex]
6. Calculate Future Amount: - Use the future amount formula [tex]\( A = I \times (1 + r)^n \)[/tex] [tex]\[ A = 150 \times (1 + 0.73)^4 \][/tex]
7. Performing the Calculation: - First calculate [tex]\( 1 + r \)[/tex]: [tex]\[ 1 + 0.73 = 1.73 \][/tex]
- Next, raise this to the power of [tex]\( n = 4 \)[/tex]: [tex]\[ 1.73^4 = 8.95745 \][/tex]
- Finally, multiply the result by the initial amount [tex]\( I \)[/tex]: [tex]\[ A = 150 \times 8.95745 = 1343.6175615 \][/tex]
Therefore, after 8 days, the bacteria colony would grow to approximately 1343.6175615 microorganisms.
So, the future amount is approximately 1343.62 microorganisms when rounded to two decimal places.
