To determine which function correctly represents the scenario of the bacteria population increasing at a rate of [tex]\(115\%\)[/tex] each hour, we need to analyze what each function is representing mathematically and compare it to the situation provided.
1. The initial population of the bacteria colony is 1500.
2. The population grows by [tex]\(115\%\)[/tex] each hour. This means the population after each hour is 115% of the population in the previous hour.
Let's break down the given options:
### Option 1: [tex]\( f(x) = 1500(1.15)^x \)[/tex]
- Here, [tex]\(1500\)[/tex] is the initial population.
- The 115% growth rate means the population increases to [tex]\(100\% + 15\% = 115\%\)[/tex] of its previous value every hour.
- In decimal form, [tex]\(115\%\)[/tex] is equivalent to multiplying by [tex]\(1.15\)[/tex].
- Thus, after [tex]\(x\)[/tex] hours, the population would be [tex]\(1500 \cdot (1.15)^x\)[/tex].
### Option 2: [tex]\( f(x) = 1500(115)^x \)[/tex]
- Again, [tex]\(1500\)[/tex] is the initial population.
- However, if you multiply the population by [tex]\(115\)[/tex], it represents a 11500% increase ([tex]\(115^x\)[/tex]) every hour, which is obviously far too high for a 115% growth rate.
### Option 3: [tex]\( f(x) = 1500(2.15)^x \)[/tex]
- Here, [tex]\(2.15\)[/tex] represents a 215% increase ([tex]\(100\% + 115\% = 215\%\)[/tex]).
- It incorrectly suggests that the population increases to [tex]\(215\%\)[/tex] of its previous value each hour rather than [tex]\(115\%\)[/tex].
### Option 4: [tex]\( f(x) = 1500(215)^x \)[/tex]
- Here, [tex]\(215\)[/tex] represents 21500% growth ([tex]\(215^x\)[/tex]) per hour, widely overestimating the population growth rate.
Based on these analyses, the correct function to represent the growth of the bacteria colony at a [tex]\(115\%\)[/tex] increase per hour is:
[tex]\[ f(x) = 1500(1.15)^x \][/tex]
