Question

exemplar 2
consider the function for population growth. the population doubles every year.
\\( p(x) = 120(2)^x \\)
what do the values 120 and 2 represent in relation to the population?
a. the 120 is the initial population, and the 2 indicates that the population doubles every year.
b. the 120 is the initial population, and the 2 indicates that the population doubles every year.
c. the 120 is the initial population, and the 2 indicates that the population increases by 2 every 2 years.
d. the 120 is the population at 1 year, and the 2 indicates that the population increases by 2 every year.

Explanation:

The function for population growth is given by \( P(x) = 120(2)^x \), where \( x \) represents the number of years. In an exponential growth function of the form \( P(x)=a(b)^x \), \( a \) is the initial amount (when \( x = 0 \)), and \( b \) is the growth factor. When \( x = 0 \), \( P(0)=120(2)^0 = 120\times1=120 \), so 120 is the initial population. The base of the exponent, 2, indicates that the population multiplies by 2 (doubles) each year as \( x \) (number of years) increases by 1.

Now let's analyze each option:

• Option A: Matches our analysis. 120 is initial population, 2 means population doubles yearly.
• Option B: Same as A? Wait, no, wait the options: Wait, option A and B? Wait no, looking back, option A and B? Wait the original options: Wait the user's options: A says 120 is initial, 2 indicates doubles every year. B says same? Wait no, maybe a typo? Wait no, let's recheck. Wait the options:

Wait the user's options:

A. The 120 is the initial population, and the 2 indicates that the population doubles every year.

B. The 120 is the initial population, and the 2 indicates that the population increases by 2 every 2 years. (No, 2 is a growth factor for doubling, not increasing by 2 every 2 years)

C. The 120 is the initial population, and the 2 indicates that the population increases by 2 every 2 years. (Incorrect, 2 is for doubling yearly)

D. The 120 is the population at 1 year, and the 2 indicates that the population increases by 2 every year. (When \( x = 1 \), \( P(1)=120\times2 = 240 \), so 120 is not population at 1 year, and 2 is not for increasing by 2 yearly)

So the correct option is A.

Answer:

A. The 120 is the initial population, and the 2 indicates that the population doubles every year.