To determine the correct function representing the growth of the bacteria colony, we need to analyze the given information:
- The initial number of bacteria is 1500.
- The population increases at a rate of [tex]\( 115\% \)[/tex] each hour.
- [tex]\( x \)[/tex] represents the number of hours elapsed.
Since the population increases by [tex]\( 115\% \)[/tex] every hour, this means that each hour, the population is [tex]\( 115\% \)[/tex] of its current population in addition to the initial population. Mathematically speaking:
1. Converting the percentage rate to a decimal:
[tex]\[
115\% = 1.15
\][/tex]
2. The population each hour is multiplied by a factor of [tex]\( 1.15 \)[/tex].
To express this growth, we use an exponential growth function of the form:
[tex]\[
f(x) = P_0 (\text{rate})^x
\][/tex]
Where [tex]\( P_0 \)[/tex] is the initial population, the rate is the growth factor per unit time, and [tex]\( x \)[/tex] is the time in hours.
Substituting the given values (initial population [tex]\( P_0 = 1500 \)[/tex] and growth rate [tex]\( 1.15 \)[/tex]) into the function, we get:
[tex]\[
f(x) = 1500(1.15)^x
\][/tex]
Now, let's compare this function to the given options:
1. [tex]\( f(x) = 1500(1.15)^x \)[/tex]
2. [tex]\( f(x) = 1500(115)^x \)[/tex]
3. [tex]\( f(x) = 1500(2.15)^x \)[/tex]
4. [tex]\( f(x) = 1500(215)^x \)[/tex]
The correct representation is option 1:
[tex]\[
f(x) = 1500(1.15)^x
\][/tex]
Therefore, the function that correctly represents the scenario where the bacteria population increases by [tex]\( 115\% \)[/tex] each hour is:
[tex]\[
\boxed{f(x) = 1500(1.15)^x}
\][/tex]
