Question

a new tennis club opened with 25 members. the membership of the club can be approximated by the formula c = 25(1.2)^m where m represents the number of months and c represents the total club membership. after how many months will the club have about 43 members?
a 2
b 3
c 4
d 5

Explanation:

Step1: Set up the equation

We know the formula \( C = 25(1.2)^m \), and we want to find \( m \) when \( C = 43 \). So we set up the equation \( 43=25(1.2)^m \).

Step2: Divide both sides by 25

Divide both sides of the equation by 25: \( \frac{43}{25}=(1.2)^m \), which simplifies to \( 1.72=(1.2)^m \).

Step3: Take logarithm on both sides

Take the natural logarithm (or common logarithm) of both sides. Using natural logarithm: \( \ln(1.72)=m\ln(1.2) \).

Step4: Solve for m

Solve for \( m \) by dividing both sides by \( \ln(1.2) \): \( m = \frac{\ln(1.72)}{\ln(1.2)} \). Calculate the values: \( \ln(1.72)\approx0.5423 \), \( \ln(1.2)\approx0.1823 \). Then \( m\approx\frac{0.5423}{0.1823}\approx3 \) (we can also check by plugging in \( m = 3 \): \( 25(1.2)^3=25\times1.728 = 43.2\approx43 \)).

Answer:

B. 3