The manager of a health club determines that the club's membership has increased at a rate of 16 percent per year for the past four years. The club currently has 42 members. If this trend continues, how many years will it take for the club's membership to exceed 100 membgers ?

Options

A. 4 years
B. 5 years
C. 6 years
D. 7 years

Answer

To solve the problem of how many years it will take for the health club's membership to exceed 100 members given a current membership of 42 and an annual growth rate of 16%, we can use the concept of exponential growth.

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- Current membership (P) = 42 members
- Growth rate (r) = 16% = 0.16
- We want to find the number of years (t) it takes for the membership to exceed 100 members.

2. **Set Up the Exponential Growth Formula**:
The formula for exponential growth can be expressed as:
\[ P(t) = P_0 \cdot (1 + r)^t \]
where:
- \( P(t) \) is the future population (membership),
- \( P_0 \) is the initial population (42 members),
- \( r \) is the growth rate (0.16),
- \( t \) is the number of years.

3. **Substitute the Known Values**:
We want to find \( t \) such that:
\[ 42 \cdot (1 + 0.16)^t > 100 \]
This simplifies to:
\[ 42 \cdot (1.16)^t > 100 \]

4. **Divide Both Sides by 42**:
\[ (1.16)^t > \frac{100}{42} \]
Calculating the right side:
\[ \frac{100}{42} \approx 2.38 \]

5. **Take the Logarithm of Both Sides**:
To solve for \( t \), we take the logarithm:
\[ \log((1.16)^t) > \log(2.38) \]
Using the power rule of logarithms:
\[ t \cdot \log(1.16) > \log(2.38) \]

6. **Solve for \( t \)**:
\[ t > \frac{\log(2.38)}{\log(1.16)} \]
Now, calculating the logarithms:
- \( \log(2.38) \approx 0.376 \)
- \( \log(1.16) \approx 0.063 \)

Therefore:
\[ t > \frac{0.376}{0.063} \approx 5.95 \]

7. **Interpret the Result**:
Since \( t \) must be a whole number (you can't have a fraction of a year in this context), we round up to the next whole number. Thus:
\[ t = 6 \]

### Conclusion:
It will take approximately **6 years** for the club's membership to exceed 100 members.