Exponential functions

Part A

Bacteria grow exponentially. There is often a doubling time associated with this growth. A doubling time is the amount of time it takes for the population to double.

In the first part, you will take on the role of a scientist studying bacteria. The bacteria growing in a petri dish is exponentially modelled by the equation:

A(t) = 16(2)^t - 6, where A is the number of bacteria and t is time in hours.

A scientist has injected the bacteria with a poison that decays in the bacteria exponentially.

The amount of poison in the bacteria colony is modelled by the equation:

P(t) = 100(0.5)^t, Where P is the percentage of poison remaining and t is the time in hours.

1. Graph both functions on the same grid with the vertical axis representing both the number of bacteria, A, and the percentage of poison, P. You can graph using hand. Include titles, units and scales, and ensure you label each function.

2. Based on both graphs state:

A. The y-intercepts.

B. Whether the function is decreasing or increasing from left to right and explain how you can tell based on the equations.

3. Algebraically determine the number of bacteria and the percentage of poison remaining after 30 minutes.

4. Estimate when there will be 20 bacteria.

A. Graphically.

B. Algebraically and with trial and error to one decimal place.

5. Algebraically and with trial and error determine when there will be 30% poison left to one decimal place.

6. Solve 16(2)^t - 6 = 100(0.5)^t using the graph and confirm using a left side/right side table.

Part B

In this part, you will be analyzing the growth of internet usage since 1995. It can be modelled by the function:

A(t) = [5 (1.08) ^t-5] [2 (1.08)^t]². Where A is the number of users in millions and t is the number of years since 2000.

1. Simplify the equation using exponent laws.

2. Find the number of internet users after 3 years and 3 months, or 13/4 years. Solve by simplifying, then converting to a radical.

3. Using the simplified equation from 1), approximately how many internet users will there be in 2019?

4. In which year will there be 1 billion users?

A. Estimate using a graph. Note that the graph can be created by hand.

B. Estimate using trial and error.

C. Solve exactly using logarithms.

5. Explain some pros and cons for each option in Question 4.