Question

Country A and country B currently have the same population but the population of Country A is growing at a rate of 5% per year but the population of Country B is growing at a rate of 21.551% per year. What expression represents the ratio, when converted to a decimal, of Country A's population to Country B's population after x years? (1.05^4=1.21551)

Answer 1

Answer: $f(x)= (0.863834933)^x$

Step-by-step explanation:

Let the initial  population of country A represented by $P_1$ and the population of country B is represented by $P_2$,

Then, According to the question,

$P_1=P_2$ ----------(1)

Since, the population of country A is increasing at a rate of 5% per year,

Hence, the population of A after x years

$= P_1(1+0.05)^x$

$= P_1(1.05)^x$

Similarly,  the population of country B is increasing at a rate of 21.551 % per year,

Hence, the population of B after x years

$= P_2(1+0.21551)^x$

$= P_2(1.21551)^x$

Thus, the ratio of the population of A and that of B is,

$\frac{\text{ Population of A}}{\text{Population of B}}=\frac{ P_1(1.05)^x}{ P_2(1.21551)^x}$

By equation (1),

$\frac{\text{ Population of A}}{\text{Population of B}}=\frac{ P_1(1.05)^x}{ P_1(1.21551)^x}$

$\frac{\text{ Population of A}}{\text{Population of B}}=\frac{ (1.05)^x}{ (1.21551)^x}$

$\frac{\text{ Population of A}}{\text{Population of B}}=(\frac{1.05}{1.21551})^x$

$f(x)= (0.863834933)^x$

Which is the required function.