Question

consider the function for cell reproduction. the cells duplicate every minute.
$f(x) = 30(2)^x$
what do the values 30 and 2 represent in relation to the duplication of these cells?
a. the 30 is the initial number of cells, and the 2 indicates that the number of cells doubles every minute.
b. the 30 is the initial number of cells, and the 2 indicates that the number of cells increases by 2 every 2 minutes.
c. the 30 is the initial number of cells, and the 2 indicates that the number of cells increases by 2 every minute.
d. the 30 is the initial number of cells at 1 minute, and the 2 indicates that the number of cells increases by 2 every minute.

Explanation:

The function for cell reproduction is given by \( f(x) = 30(2)^x \), which is an exponential growth function of the form \( f(x)=a(b)^x \), where \( a \) is the initial amount, \( b \) is the growth factor, and \( x \) is the time.

• For \( a = 30 \): When \( x = 0 \) (initial time, at \( t = 0 \) minutes), \( f(0)=30(2)^0=30\times1 = 30 \). So 30 represents the initial number of cells.
• For \( b = 2 \): Since the cells duplicate every minute, the number of cells at each minute is twice the number of cells in the previous minute (e.g., at \( x = 1 \), \( f(1)=30\times2 \); at \( x = 2 \), \( f(2)=30\times2^2 \), etc.). So 2 indicates that the number of cells doubles every minute.

Now let's analyze each option:

• Option A: Matches our analysis. 30 is the initial number of cells, and 2 indicates doubling every minute.
• Option B: The 2 does not indicate an increase by 2 every 2 minutes. The function shows doubling (multiplying by 2) every minute, not an additive increase over 2 minutes.
• Option C: The 2 does not indicate an increase by 2 every minute. It's a multiplicative factor (doubling), not an additive increase of 2.
• Option D: At \( x = 0 \) (initial time, not at 1 minute), we get 30, so 30 is the initial number of cells at \( x = 0 \), not at 1 minute. Also, the 2 is for doubling, not an additive increase of 2.

Answer:

A. The 30 is the initial number of cells, and the 2 indicates that the number of cells doubles every minute.