To determine the monthly growth rate from the expression [tex]\( f(x) = 12(1.035)^x \)[/tex], we carefully analyze the given function.
The given function models the growth of the membership with respect to the variable [tex]\( x \)[/tex], which typically represents time in months in such contexts.
The general form of an exponential growth function is:
[tex]\[ f(x) = a(b)^x \][/tex]
- [tex]\( a \)[/tex] represents the initial amount or starting value.
- [tex]\( b \)[/tex] is the growth factor.
In our given function:
[tex]\[ f(x) = 12(1.035)^x \][/tex]
- Here, the initial membership ([tex]\( a \)[/tex]) is 12 members.
- The growth factor ([tex]\( b \)[/tex]) is 1.035.
To find the growth rate, we need to understand what this growth factor means. The growth factor [tex]\( b \)[/tex] can be described in the following way:
[tex]\[ b = 1 + r \][/tex]
where [tex]\( r \)[/tex] is the growth rate in decimal form. Solving for [tex]\( r \)[/tex]:
[tex]\[ 1 + r = 1.035 \][/tex]
[tex]\[ r = 1.035 - 1 \][/tex]
[tex]\[ r = 0.035 \][/tex]
To convert this decimal form to a percentage, we multiply by 100:
[tex]\[ r \times 100 = 0.035 \times 100 = 3.5\% \][/tex]
Thus, the monthly growth rate for the drama club membership is [tex]\( 3.5\% \)[/tex].
Hence, the correct answer is:
C. [tex]\( 3.5\% \)[/tex]
