To determine the future size of a bacteria colony that increases by 67% every 2 days, we need to use the compound interest formula:
[tex]\[ \text{Future Amount} = I(1 + r)^t \][/tex]
Let's break down the problem step-by-step:
1. Initial Number of Bacteria (I): The colony initially has 55 bacteria microorganisms.
[tex]\[ I = 55 \][/tex]
2. Rate of Increase (r): The bacteria colony increases by 67% every 2 days. Converting this percentage to a decimal gives us:
[tex]\[ r = 0.67 \][/tex]
3. Time Period (t): We want to find the size of the colony after 10 days. Since the rate is given for every 2 days, we need to calculate the number of 2-day periods in 10 days:
[tex]\[ t = \frac{10 \, \text{days}}{2 \, \text{days/period}} = 5 \][/tex]
Now, substitute these values into the formula:
[tex]\[ \text{Future Amount} = 55 \times (1 + 0.67)^5 \][/tex]
Calculate the expression inside the parentheses first:
[tex]\[ 1 + 0.67 = 1.67 \][/tex]
Next, raise this to the power of 5:
[tex]\[ 1.67^5 \approx 12.989 \][/tex]
Now multiply this by the initial number of bacteria:
[tex]\[ 55 \times 12.989 \approx 714.4059208384998 \][/tex]
Therefore, the size of the bacteria colony after 10 days will be approximately 714.41 microorganisms.
So the Future Amount is approximately:
[tex]\[ 714.41 \, \text{microorganisms} \][/tex]
