Question

A scientist is studying the population growth rates of two different species of rats in the same laboratory conditions. The population of species A is represented by the equation below, where P represents the population at the end of x weeks.

P = 2^x + 19

The population of species B is represented by the equation below, where P represents the population at the end of x weeks.

P = 3^x + 18

Which statement is true?

A. At the end of 21 weeks, species A and species B will have the same population.
B. At the end of 18 weeks, species A and species B will have the same population.
C. At the end of 3 weeks, species A and species B will have the same population.
D. At the end of 1 week, species A and species B will have the same population.

Answer 1

At the end of 1 week, species A and species B will have the same population.

Answer 2

Answer:

Option (D) is correct

Step-by-step explanation:

According to the problem given,

$P= 2^{x}+19\:---(1)$

$and\\ P= 3^{x}+18\:---(2)$

/* From (1) and (2),

$2^{x}+19=3^{x}+18$

$\implies 2^{x}-3^{x}+19-18=0$

$\implies 2^{x}-3^{x}+1=0\:---(3)$

Case 1:

if x = 21 ,

$2^{21}-3^{21}+1=0 \:(False)$

Case 2:

If x = 18 ,

$2^{18}-3^{18}+1=0 \:(False)$

Case 3:

If x = 3 ,

$2^{3}-3^{3}+1=0 \:(False)$

Case 4:

If x = 1,

$2^{1}-3^{1}+1=0 \:(True)$

Therefore,

Option (D) is true

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