To determine which function represents the population growth of a bacteria colony given the initial population and the growth rate, we need to carefully analyze the conditions provided and use the exponential growth model.
Here's a step-by-step breakdown:
1. Initial Population: The population starts with 1500 bacteria. This value is the initial amount and forms the base of our growth model.
2. Growth Rate: The population increases by 115% each hour. To express growth rate in a multiply form suitable for exponential functions:
- 100% of the current population means the population remains the same.
- An increase by 115% means the new population will be 100% + 15% of the current population which is 1 + 0.15 = 1.15 times the current population.
3. Exponential Growth Formula: For modeling population growth, exponential functions are used. The general form of an exponential growth function is:
[tex]\[
f(x) = P_0 \cdot (1 + r)^x
\][/tex]
where [tex]\(P_0\)[/tex] is the initial population, [tex]\(r\)[/tex] is the growth rate, and [tex]\(x\)[/tex] is the time period (in this case, hours).
4. Constructing the Function: Given the initial population [tex]\(P_0 = 1500\)[/tex] and the growth rate [tex]\(r = 1.15\)[/tex], we can substitute these values into the formula:
[tex]\[
f(x) = 1500 \cdot (1.15)^x
\][/tex]
Therefore, the function that accurately represents this scenario is:
[tex]\[
f(x) = 1500(1.15)^x
\][/tex]
Comparing this to the choices given:
- [tex]\(f(x) = 1500(1.15)^x\)[/tex] matches our constructed function.
The others:
- [tex]\(f(x) = 1500(115)^x\)[/tex] suggests the population grows by a factor of 115 each hour, which is incorrect.
- [tex]\(f(x) = 1500(2.15)^x\)[/tex] suggests the population more than doubles each hour, which is incorrect.
- [tex]\(f(x) = 1500(215)^x\)[/tex] similarly suggests an unrealistic, extremely high growth rate.
Thus, the correct choice is:
[tex]\[ f(x)=1500(1.15)^x \][/tex]
