Respuesta :
Hello there. To solve this question, we'll have to remember some properties about population growth.
In 1990, Jamaica's population of 2466000 was expected to grow exponentially by 1.1% each year. We want to determine the range of the exponential function.
First, remember that for exponential growth of a population, given the initial population P0, the rate of growth r, we can set a function P(t) that related population P and time t as follows:
[tex]P(t)=P_0\cdot e^{rt}[/tex]In which we can plug in the values P0 = 2466000, r = 0.011 (after converting to decimals) to get:
[tex]P(t)=2466000\cdot e^{0.011t}[/tex]Knowing that:
[tex]e^x>0,\forall x\in\mathfrak{\Re }[/tex]It is easy to show that neither All real numbers and y >= 0 can't be an option, since e^(0.011t) is greater than zero, all real numbers would contain the negative ones. y >= 0 contains the point zero, which is also not true for the function.
The only options left are y >= 2466000 or x >= 0.
In this case, we didn't say that e^x >= 1, therefore to conclude that y >= 2466000. In fact, for x < 0, we have that 0 < e^x < 0 and then y would not be equal or greater than 2466000.
This means that x >= 0 is the answer. This happens because we only take count of time from when we started the observation, namely 1990 as the origin.
