5. After Mrs. Albee finally relents and agrees to make a TikTok account, where she only posts videos of her telling mat jolkes. As you can imagine, the number of people that follow her account rapidly grows! The number of followers for Mrs, Albee's TikTok account is modeled by the function F . The number of followers is expected to increase by $$ 31.4 \% $$ each week. At time $$ t=0 $$ days, Mrs. Albee had 100 followers. a) If $$ t $$ is measured in days, what is an expression for $$ F(t) $$ (Note: There are seven days in a week) b) The most popular account on TikTolk has 156 million followers. According to the model found in part a, how many days will tr take for Mrs. Albee to have the most followers on TikTok (Hint: Use a guess and check method)?

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To solve the problem, we will first derive the expression for the number of followers $$ F(t) $$ based on the information provided.

### Part a: Expression for $$ F(t) $$

Given that Mrs. Albee's followers increase by $$ 31.4\% $$ each week, we can express this as a growth factor. The growth factor for a $$ 31.4\% $$ increase is:

$$
1 + 0.314 = 1.314
$$

Since $$ t $$ is measured in days and there are 7 days in a week, we can express the number of weeks as $$ \frac{t}{7} $$. The number of followers after $$ t $$ days can be modeled by the exponential growth function:

$$
F(t) = F(0) \cdot (1.314)^{\frac{t}{7}}
$$

Given that $$ F(0) = 100 $$ followers, we can substitute this into the equation:

$$
F(t) = 100 \cdot (1.314)^{\frac{t}{7}}
$$

### Part b: Finding the time $$ t $$ when $$ F(t) $$ reaches 156 million followers

We need to find $$ t $$ such that:

$$
F(t) = 156,000,000
$$

Substituting the expression for $$ F(t) $$:

$$
100 \cdot (1.314)^{\frac{t}{7}} = 156,000,000
$$

Dividing both sides by 100:

$$
(1.314)^{\frac{t}{7}} = 1,560,000
$$

Next, we take the logarithm of both sides to solve for $$ t $$:

$$
\frac{t}{7} \cdot \log(1.314) = \log(1,560,000)
$$

Now, we can isolate $$ t $$:

$$
t = 7 \cdot \frac{\log(1,560,000)}{\log(1.314)}
$$

Now we can calculate the values of the logarithms:

1. Calculate $$ \log(1,560,000) $$:
   - Using a calculator, $$ \log(1,560,000) \approx 6.193 $$

2. Calculate $$ \log(1.314) $$:
   - Using a calculator, $$ \log(1.314) \approx 0.119 $$

Now substituting these values into the equation for $$ t $$:

$$
t = 7 \cdot \frac{6.193}{0.119} \approx 7 \cdot 52.0 \approx 364
$$

Thus, it will take approximately **364 days** for Mrs. Albee to have the most followers on TikTok.
a) $$ F(t) = 100 \cdot (1.314)^{\frac{t}{7}} $$ b) It will take approximately 364 days for Mrs. Albee to have the most followers on TikTok.

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