Question 1: Write the general formula for exponential growth and decay. Identify the key variables involved.

Answer: y = a(1 + r)^t for growth, y = a(1 - r)^t for decay, where a = initial amount, r = rate, t = time

Question 2: Calculate the amount after 5 years if $1000 decays at an annual rate of 5%.

Answer: 1000 * (1 - 0.05)^5 ≈ 773.78

Question 3: A bacteria population doubles every 3 hours. If there are initially 500 bacteria, how many will there be after 9 hours?

Answer: 500 * 2^(9/3) = 500 * 2^3 = 500 * 8 = 4000

Question 4: Plot the graph of y = 200(1 + 0.1)^t for t from 0 to 10 on the grid. Describe the trend.

Answer: The graph shows exponential growth; the amount increases rapidly over time.

Question 5: A radioactive substance decays so that only 30% remains after 4 hours. Find the decay rate per hour.

Answer: 0.3 = (1 - r)^4; r = 1 - (0.3)^(1/4) ≈ 1 - 0.737 = 0.263 or 26.3%

Question 6: A population of 200 increases by 8% annually. What will be the population after 3 years?

Answer: 200 * (1 + 0.08)^3 ≈ 200 * 1.2597 ≈ 251.94

Question 7: Construct a graph to show exponential decay starting from 100 and decreasing at 10% per year. Label the axes.

Answer: Graph shows a decreasing exponential curve; y = 100(1 - 0.10)^t

Question 8: Identify the common mistake in the following decay calculation: Using y = 100 * (1 - 0.10)^t for t=5, getting 59.05. Is this correct? Why or why not?

Answer: Yes, because it correctly applies the decay formula with r=10%.

Question 9: A car’s value depreciates by 15% each year. If its original value was $20,000, what is its value after 4 years?

Answer: 20000 * (1 - 0.15)^4 ≈ 20000 * 0.522 = $10,440

Question 10: A savings account grows by 6% annually. How long will it take for the balance to double if the initial amount is $5000?

Answer: Solve 2 = (1 + 0.06)^t; t = log(2)/log(1.06) ≈ 11.89 years

Question 11: Explain why exponential decay models are suitable for radioactive substances but not for constant-rate decline like depreciation.

Answer: Radioactive decay is proportional to the current amount, fitting an exponential decay. Constant-rate decline implies linear decrease, not exponential.