A biologist is studying the growth of a bacterial colony. Initially, the colony has 100 bacteria. After 1 hour, the colony has 200 bacteria, and after 2 hours, it has 400 bacteria. Describe which type of model (linear, quadratic, or exponential) best fits this data and explain why.

A car rental company charges a base fee of $50 plus $0.20 per mile driven. Write an equation to model the total cost $C$ as a function of the number of miles $m$ driven. Identify the type of model used and justify your choice.

A ball is thrown into the air, and its height $h$ in meters after $t$ seconds is given by the equation $h=-5t^2+20t+1$. Identify the type of model used and explain why it is appropriate for this scenario.

A population of rabbits in a closed ecosystem grows according to the equation $P(t)=100\cdot 2^t$, where $P(t)$ is the population at time $t$ in years. Explain why this model is appropriate and identify the type of model used.

A company's revenue $R$ in thousands of dollars over time $t$ in years is modeled by the equation $R(t)=5t^2+10t+20$. Identify the type of model used and explain why it is appropriate for this scenario.

Explain how residuals can be used to assess the fit of a linear regression model.

How can you use the coefficient of determination $R^2$ to evaluate the fit of a regression model?

What is the purpose of using a scatter plot when determining the best-fitting model for a dataset?

How can you determine if a logarithmic model is a good fit for a dataset?

Explain why it is important to consider the initial conditions when selecting an exponential model for population growth.

Describe a scenario where a logarithmic model might be more appropriate than an exponential model.

Why might an exponential model be inappropriate for predicting long-term economic growth?

How can you validate the assumption that a given data set follows an exponential trend?

Discuss the importance of the base of the exponential function when modeling real-world problems.