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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)

Respuesta :

Answer: [tex]f(x)= (0.863834933)^x[/tex]

Step-by-step explanation:

Let the initial  population of country A represented by [tex]P_1[/tex] and the population of country B is represented by [tex]P_2[/tex],

Then, According to the question,

[tex]P_1=P_2[/tex] ----------(1)

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

Hence, the population of A after x years

[tex]= P_1(1+0.05)^x[/tex]

[tex]= P_1(1.05)^x[/tex]

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

Hence, the population of B after x years

[tex]= P_2(1+0.21551)^x[/tex]

[tex]= P_2(1.21551)^x[/tex]

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

[tex]\frac{\text{ Population of A}}{\text{Population of B}}=\frac{ P_1(1.05)^x}{ P_2(1.21551)^x}[/tex]

By equation (1),

[tex]\frac{\text{ Population of A}}{\text{Population of B}}=\frac{ P_1(1.05)^x}{ P_1(1.21551)^x}[/tex]

[tex]\frac{\text{ Population of A}}{\text{Population of B}}=\frac{ (1.05)^x}{ (1.21551)^x}[/tex]

[tex]\frac{\text{ Population of A}}{\text{Population of B}}=(\frac{1.05}{1.21551})^x[/tex]

[tex]f(x)= (0.863834933)^x[/tex]

Which is the required function.

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