Question: After Mr. Passwater finally relents and agrees to make a TikTok account, where he only posts videos of him singing math jolee. As you can imagine, the number of people that follow his account rapidly grows! The number of followers for Mr. [redacted]’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, Mr Passwater 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) Y = .314(100)^x b) The most popular account on TikTok has 1.5 million followers. Assuming the model found in part a. how many days will it take for Mr. [redacted] to have the most followers on TikTok? (Hint: Use a graph and check manually) Ivins 6. The number of bacteria found in a petri dish can be modeled by the function B(t) = kt, where k is a constant and t is measured in hours. At time t = 2.5, there were 5,100 bacteria in the petri dish. According to this model, what is the number of bacteria in the petri dish after 90 minutes? 7. Mr. [redacted] decides to buy a brand new 2024 MINI Cooper convertible for $29,415. The value of the car is expected to depreciate (decrease) by 17% each year for the first five years. Let V represent the value of the brand new MINI Cooper at time t years after purchase. a) Write an expression for V(t) where V(0) = a Ivins b) Let V(m) = a^x be an equivalent expression for V(m) where m is the number of months after purchase. Find the value of x. Worksheet A: Topic 2.5 Exponential Functions, Growth and Decay Modeling General Math 2 Precalculus

1. **Problem Statement:** We are given that the number of followers for Mr. Passwater's TikTok account grows by 31.4% each week, starting with 100 followers at time $t=0$ days. We want to find: a) An expression for $F(t)$ where $t$ is in days. b) How many days it will take for Mr. Passwater to reach 1.5 million followers. 2. **Formula for Exponential Growth:** The general formula for exponential growth is: $$F(t) = F_0 \times (1 + r)^n$$ where: - $F_0$ is the initial amount, - $r$ is the growth rate per period, - $n$ is the number of periods. 3. **Part a) Expression for $F(t)$:** - Growth rate per week: $r = 0.314$ - Initial followers: $F_0 = 100$ - Since $t$ is in days and growth is weekly, number of weeks is $\frac{t}{7}$. So, $$F(t) = 100 \times (1 + 0.314)^{\frac{t}{7}} = 100 \times (1.314)^{\frac{t}{7}}$$ 4. **Part b) Days to reach 1.5 million followers:** Set $F(t) = 1,500,000$ and solve for $t$: $$1,500,000 = 100 \times (1.314)^{\frac{t}{7}}$$ Divide both sides by 100: $$\frac{1,500,000}{100} = (1.314)^{\frac{t}{7}}$$ $$15,000 = (1.314)^{\frac{t}{7}}$$ Take natural logarithm on both sides: $$\ln(15,000) = \ln\left((1.314)^{\frac{t}{7}}\right) = \frac{t}{7} \ln(1.314)$$ Solve for $t$: $$t = 7 \times \frac{\ln(15,000)}{\ln(1.314)}$$ Calculate values: - $\ln(15,000) \approx 9.6158$ - $\ln(1.314) \approx 0.2731$ So, $$t \approx 7 \times \frac{9.6158}{0.2731} = 7 \times 35.22 = 246.54$$ **Answer:** It will take approximately 247 days for Mr. Passwater to reach 1.5 million followers. --- **Note:** The other problems (6 and 7) are not solved here as per instructions to solve only the first question.