1. 9.6 – Exponential Growth and Decay; Modeling Data
Math 71B
9.6 – Exponential Growth and Decay; Modeling Data

2. Population growth/decline can be modeled by the function: 𝑨= 𝑨 𝟎 𝒆 𝒌𝒕 𝐴 is the amount of people at time 𝑡 𝐴 0 is the amount of people at time 𝑡=0 𝑘 is a constant representing the growth rate (% increase/decrease each year over the last year)

3. Population growth/decline can be modeled by the function: 𝑨= 𝑨 𝟎 𝒆 𝒌𝒕 𝐴 is the amount of people at time 𝑡 𝐴 0 is the amount of people at time 𝑡=0 𝑘 is a relative growth rate (ex: 0.02 would represent 2% of the population at time 𝑡)

4. Population growth/decline can be modeled by the function: 𝑨= 𝑨 𝟎 𝒆 𝒌𝒕 𝐴 is the amount of people at time 𝑡 𝐴 0 is the amount of people at time 𝑡=0 𝑘 is a relative growth rate (ex: 0.02 would represent 2% of the population at time 𝑡)

5. Population growth/decline can be modeled by the function: 𝑨= 𝑨 𝟎 𝒆 𝒌𝒕 𝐴 is the amount of people at time 𝑡 𝐴 0 is the amount of people at time 𝑡=0 𝑘 is a relative growth rate (ex: 0.02 would represent 2% of the population at time 𝑡)

6. Ex 1. Suppose 𝐴=4.1 𝑒 0.01𝑡 represents the population in New Zealand (in millions) 𝑡 years after What was the population of New Zealand in 2006? 4.1 million What is New Zealand’s growth rate? 0.01 = 1% When will New Zealand’s population be 8 million?

7. Ex 1. Suppose 𝐴=4.1 𝑒 0.01𝑡 represents the population in New Zealand (in millions) 𝑡 years after What was the population of New Zealand in 2006? 4.1 million What is New Zealand’s growth rate? 0.01 = 1% When will New Zealand’s population be 8 million?

8. Ex 1. Suppose 𝐴=4.1 𝑒 0.01𝑡 represents the population in New Zealand (in millions) 𝑡 years after What was the population of New Zealand in 2006? 4.1 million What is New Zealand’s growth rate? 0.01 = 1% per year When will New Zealand’s population be 8 million?

9. Ex 1. Suppose 𝐴=4.1 𝑒 0.01𝑡 represents the population in New Zealand (in millions) 𝑡 years after What was the population of New Zealand in 2006? 4.1 million What is New Zealand’s growth rate? 0.01 = 1% When will New Zealand’s population be 8 million?

10. Ex 1. Suppose 𝐴=4.1 𝑒 0.01𝑡 represents the population in New Zealand (in millions) 𝑡 years after What was the population of New Zealand in 2006? 4.1 million What is New Zealand’s growth rate? 0.01 = 1% When will New Zealand’s population be 8 million?

11. Ex 2. What’s different about a population modeled by the formula 𝐴=142
Ex 2. What’s different about a population modeled by the formula 𝐴=142.9 𝑒 −0.004𝑡 , and what does it mean? The growth rate is negative. This means that the population is declining.

12. Ex 2. What’s different about a population modeled by the formula 𝐴=142
Ex 2. What’s different about a population modeled by the formula 𝐴=142.9 𝑒 −0.004𝑡 , and what does it mean? The growth rate (𝒌) is negative. (−𝟎.𝟒% here) This means that the population is declining.

13. Ex 2. What’s different about a population modeled by the formula 𝐴=142
Ex 2. What’s different about a population modeled by the formula 𝐴=142.9 𝑒 −0.004𝑡 , and what does it mean? The growth rate (𝒌) is negative. (−𝟎.𝟒% here) This means that the population is declining.

14. Ex 2. What’s different about a population modeled by the formula 𝐴=142
Ex 2. What’s different about a population modeled by the formula 𝐴=142.9 𝑒 −0.004𝑡 , and what does it mean? The growth rate (𝒌) is negative. (−𝟎.𝟒% here) This means that the population is declining.

15. Carbon-14 is a radioactive substance that exists in plants and animals
Carbon-14 is a radioactive substance that exists in plants and animals. After death, the supply in the body stops being replenished, and since the substance is radioactive, the amount of it decays over time. In fact, it decays exponentially, and can be modeled with the same 𝑨= 𝑨 𝟎 𝒆 𝒌𝒕 formula, where 𝑘<0 (so the function is decreasing).

16. The ________ of a substance is the time required for half of the given sample to disintegrate.

17. The ________ of a substance is the time required for half of the given sample to disintegrate.
half-life

18. Ex 3. Given that the half-life of carbon-14 is about 5715 years, find the exponential decay model for it.

19. Ex 4. Suppose you find a paper with ancient Greek writing on it and want to find out if it really is ancient. You do some analysis and figure out that the paper contains 95% of its original carbon-14. Estimate the age of the paper.