The number of people using a social media application is declining by 15% monthly. There are an initial 22,000,000 users. Find the half-life.

To find the half-life of the number of people using the social media application, we need to determine the time it takes for the number of users to reduce to half its initial value.

Given:
- Initial number of users, \( N_0 = 22,000,000 \)
- Monthly decline rate, \( r = 15\% = 0.15 \)

We use the exponential decay model:
\[ N(t) = N_0 \times (1 - r)^t \]

where:
- \( N(t) \) is the number of users at time \( t \),
- \( N_0 \) is the initial number of users,
- \( r \) is the decay rate per time period (month),
- \( t \) is the time in months.

We need to find \( t \) when \( N(t) = \frac{N_0}{2} \).

\[ \frac{N_0}{2} = N_0 \times (1 - r)^t \]

Dividing both sides by \( N_0 \):

\[ \frac{1}{2} = (1 - r)^t \]

Substitute \( r = 0.15 \):

\[ \frac{1}{2} = (1 - 0.15)^t \]

\[ \frac{1}{2} = (0.85)^t \]

To solve for \( t \), take the natural logarithm of both sides:

\[ \ln\left(\frac{1}{2}\right) = \ln\left((0.85)^t\right) \]

Using the property of logarithms \( \ln(a^b) = b\ln(a) \):

\[ \ln\left(\frac{1}{2}\right) = t \ln(0.85) \]

Now, solve for \( t \):

\[ t = \frac{\ln\left(\frac{1}{2}\right)}{\ln(0.85)} \]

Calculate the natural logarithms:

\[ \ln\left(\frac{1}{2}\right) = \ln(0.5) \approx -0.6931 \]
\[ \ln(0.85) \approx -0.1625 \]

So,

\[ t = \frac{-0.6931}{-0.1625} \approx 4.265 \]

Therefore, the half-life of the number of users is approximately 4.265 months.

Would you like further details or have any questions?

Here are 5 questions you might want to ask next:
1. How do you determine the decay constant for exponential decay problems?
2. Can you explain the steps to solve for the half-life in more detail?
3. How does the half-life concept apply to other fields, like chemistry or physics?
4. What happens to the number of users after 6 months at the same decline rate?
5. How would the half-life change if the decline rate were 20% instead of 15%?

**Tip:** Remember that the half-life formula can be applied to various exponential decay situations, not just social media user decline.

Learn how to calculate the half-life of social media users' decline using exponential decay and logarithms. Given an initial 22,000,000 users with a 15% monthly decline rate, find out how long it takes for the user base to halve. Suitable for high school students learning exponential decay.

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