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3. Reminder Growth and Decay We have previously examined the idea of growth when studying Compound Interest. Two questions from that presentation are shown below. Question 1 £8000 is invested at 7% compound interest for 6 years. Find: (a) the amount in the account at the end of the period (nearest £). Question 2 £1250 is invested at 9% compound interest for 10 years. Find: (a) the amount in the account at the end of the period (nearest £). Answer: 8000 x 1.07 6 = £12,006 Answer: 1250 x 1.09 10 = £2959

4. Growth Example Question 1 (b) 480 x 1.15 4 = £840 A antique dealer bought a chair for £480. If the value of the chair increases at a constant rate of 15% each year, what is the expected value of the chair (nearest £) after: (a)Three years. (b)Four years. Since 15% corresponds to 0.15, the multiplier used is 1.15 (a) 480 x 1.15 3 = £730 Growth and Decay We will look at some further examples of growth that involve the increase in value of an object over time (appreciation) rather than just the investment of money.

5. Example Question 2 (b) 185 000 x 1.08 7 = £317 000 A house purchased for £185 000 is expected to increase in value at a steady rate of 8% per annum. Find the value of the house (nearest £1000) after: (a)Four years. (b)Seven years. Since 8% corresponds to 0.08, the multiplier used is 1.08 (a) 185 000 x 1.08 4 = £252 000 Growth and Decay We will look at some further examples of growth that involve the increase in value of an object over time (appreciation) rather than just the investment of money. £185 000

6. Question 1 Question 2 Growth and Decay £205 000 A house purchased for £205 000 is expected to increase in value at a steady rate of 7% per annum. Find the value of the house (nearest £1000) after: (a)Five years. (b)Ten years. A antique dealer bought a chair for £350. If the value of the chair increases at a constant rate of 12% each year, what is the expected value of the chair (nearest £) after: (a)Four years. (b)Five years. (b) 350 x 1.12 5 = £617(a) 350 x 1.12 4 = £551 (b) 205 000 x 1.07 10 = £ 403 000(a) 205 000 x 1.07 5 = £288 000

7. Growth and Decay We will look at some further examples of growth that do not involve the investment of money or the appreciation in value of objects. (b) 800 x 1.14 8 = 2282 Example Question 3 Bacteria grown in an experiment increase at a constant rate of 14% per hour. If at the start there are 800 bacteria calculate the number after: (a)Four hours. (b)Eight hours. (a) 800 x 1.14 4 = 1351 Since 14% corresponds to 0.14, the multiplier used is 1.14

8. Growth and Decay We will look at some further examples of growth that do not involve the investment of money or the appreciation in value of objects. Example Question 4 (b) 4 250 000 x 1.02 30 = 7 698 000 The population of a small African country (4.25 million) is predicted to increase at a constant rate of 2% per year over the next fifty years. Calculate the expected population (nearest 1000) in: (a)20 years time. (b)30 years time. Since 2% corresponds to 0.02 the multiplier is 1.02 (a) 4 250 000 x 1.02 20 = 6 315 000

9. Question 3 Question 4 Growth and Decay (b) 960 x 1.04 15 = 1730(a) 960 x 1.04 6 = 1210 Bacteria grown in an experiment increase at a constant rate of 9% per hour. If at the start there are 2000 bacteria calculate the number after: (a)Five hours. (b)Nine hours. (b) 2000 x 1.09 9 = 4344(a) 2000 x 1.09 5 = 3077 The population of a small Indian Village (960 people) is predicted to increase at a constant rate of 4% per year. Calculate the expected population (nearest 10 people) of the village in: (a)6 years time. (b)15 years time.

10. Decay An example of decay can be seen when examining the loss in value (depreciation) of a new car after a number of years. (b) 12 000 x 0.82 3 = £6616 Example Question 5 A new car is bought for £12 000 and depreciates at 18% per annum. Find to the nearest £: (a)The value of the car after 2 years. (b)The value of the car after 3 years. Since the value of the car decreases the multiplier must be a fraction. And since 18% corresponds to 0.18, the multiplier is 1 – 0.18 = 0.82 (a) 12 000 x 0.82 2 = £8069 Growth and Decay

11. An example of decay can be seen when examining the loss in value (depreciation) of a new car after a number of years. Example Question 6 (b) 7500 x 0.84 4 = £3734 A new bike is bought for £7500 and depreciates at 16% per annum. Find to the nearest £: (a)The value of the bike after 3 years. (b)The value of the bike after 4 years. Since the value of the bike decreases the multiplier must be a fraction. And since 16% corresponds to 0.16, the multiplier is 1 – 0.16 = 0.84 (a) 7500 x 0.84 3 = £4445 Growth and Decay

12. Question 5 (b) 10 500 x 0.80 4 = £4301 A new car is bought for £10 500 and depreciates at 20% per annum. Find to the nearest £: (a)The value of the car after 3 years. (b)The value of the car after 4 years. (a) 10 500 x 0.80 3 = £5376 Question 6 A new bike is bought for £5400 and depreciates at 17% per annum. Find to the nearest £: (a)The value of the bike after 5 years. (b)The value of the bike after 6 years. (b) 5400 x 0.83 6 = £1765(a) 5400 x 0.83 5 = £2127 Growth and Decay

13. Example Question 7 A radioactive material decays at 5% each year. If its mass is 850g now, how much of it will there be in: (a)Six years time? (b)Ten years time? (b) 850 x 0.95 10 = 509g Since 5% corresponds to 0.05, the multiplier is 1 – 0.05 = 0.95 (a) 850 x 0.95 6 = 625g

14. Growth and Decay Example Question 8 The population of a small Island is expected to decrease steadily at a constant rate of 2% each year. If the population is currently 3600 what will it be in: (a)Twelve years time? (b)Twenty years time? (b) 3600 x 0.98 20 = 2403 Since 2% corresponds to 0.02, the multiplier is 1 – 0.02 = 0.98 (a) 3600 x 0.98 12 = 2825

15. Question 7 (b) 500 x 0.96 8 = 361g(a) 500 x 0.96 3 = 442g Question 8 (b) 2 730 000 x 0.97 30 = 1 095 000(a) 2 730 000 x 0.97 15 = 1 729 000 Growth and Decay In an experiment a 500g mass of radio active copper decays at 4% per hour. How much copper is left after: (a)Three hours? (b)Eight hours? The population of a small African country (2.73 million) is predicted to decline at a constant rate of 3% per year over the next 50 years. Calculate the expected population (nearest 1000) in: (a)Fifteen years time? (b)Thirty years time?

16. Graphs 0 Exponential Growth Curve 0 Exponential Decay Curve Growth and Decay Graphs Click  for next slide. Expect a delay.

17. Growth and Decay Graphs A culture of 800 bacteria increase at a constant rate of 14% per hour. (a)Complete the table above. (b)Draw the graph of the number of bacteria over time for the first five hours. (c)Use the graph to estimate the number of bacteria in the culture after 2½ hours. Time (hrs) 012345 Bacteria 800 0 1 2 3 4 5 Time (hours) Number of Bacteria Example Question 1 Using 1.14 as a multiplier and applying it repeatedly to table entries. 800

18. Growth and Decay Graphs A culture of 800 bacteria increase at a constant rate of 14% per hour. (a)Complete the table above. (b)Draw the graph of the number of bacteria over time for the first five hours. (c)Use the graph to estimate the number of bacteria in the culture after 2½ hours. Time (hrs) 012345 Bacteria 8009121040118513511540 0 1 2 3 4 5 Time (hours) 800 Example Question 1 Scaling the vertical axis and plotting the points. 1000 1200 1400 1600  1110 Number of Bacteria

19. Growth and Decay Graphs Time (hrs) 012345 Mass (g) 600 0 1 2 3 4 5 Time (hours) Question 1 Mass of Metal (g) In an experiment a 600g mass of radio active metal decays at a constant rate of 20% per hour. (a)Complete the table above. (b)Draw the graph of mass over time for the first five hours. (c)Use the graph to estimate how long it takes for half the mass to decay. Using 0.8 as a multiplier and applying it repeatedly to table entries.

20. Growth and Decay Graphs Time (hrs) 012345 Mass (g) 600480384307246197 0 1 2 3 4 5 Time (hours) Question 1 300 400 500 600 200 Mass of Metal (g) Scaling the vertical axis and plotting the points. In an experiment a 600g mass of radio active metal decays at a constant rate of 20% per hour. (a)Complete the table above. (b)Draw the graph of mass over time for the first five hours. (c)Use the graph to estimate how long it takes for half the mass to decay.  3 hrs 5 mins

21. Example Question 1 A antique dealer bought a chair for £480. If the value of the chair increases at a constant rate of 15% each year, what is the expected value of the chair (nearest £) after: (a)Three years. (b)Four years. Example Question 2 A house purchased for £185 000 is expected to increase in value at a steady rate of 8% per annum. Find the value of the house (nearest £1000) after: (a)Four years. (b)Seven years. Question 1 A antique dealer bought a chair for £350. If the value of the chair increases at a constant rate of 12% each year, what is the expected value of the chair (nearest £) after: (a)Four years. (b)Five years. Question 2 A house purchased for £205 000 is expected to increase in value at a steady rate of 7% per annum. Find the value of the house (nearest £1000) after: (a)Five years. (b)Ten years. Worksheet 1

22. Example Question 3 Bacteria grown in an experiment increase at a constant rate of 14% per hour. If at the start there are 800 bacteria calculate the number after: (a)Four hours. (b)Eight hours. Example Question 4 The population of a small African country (4.25 million) is predicted to increase at a constant rate of 2% per year over the next fifty years. Calculate the expected population (nearest 1000) in: (a)20 years time. (b)30 years time. Question 3 Bacteria grown in an experiment increase at a constant rate of 9% per hour. If at the start there are 2000 bacteria calculate the number after: (a)Five hours. (b)Nine hours. Question 4 The population of a small Indian Village (960 people) is predicted to increase at a constant rate of 4% per year. Calculate the expected population (nearest 10 people) of the village in: (a)6 years time. (b)15 years time. Worksheet 2

23. Example Question 5 A new car is bought for £12 000 and depreciates at 18% per annum. Find to the nearest £: (a)The value of the car after 2 years. (b)The value of the car after 3 years. Example Question 6 A new bike is bought for £7500 and depreciates at 16% per annum. Find to the nearest £: (a)The value of the bike after 3 years. (b)The value of the bike after 4 years. Question 5 A new car is bought for £10 500 and depreciates at 20% per annum. Find to the nearest £: (a)The value of the car after 3 years. (b)The value of the car after 4 years. Question 6 A new bike is bought for £5400 and depreciates at 17% per annum. Find to the nearest £: (a)The value of the bike after 5 years. (b)The value of the bike after 6 years. Worksheet 3

24. Example Question 7 A radioactive material decays at 5% each year. If its mass is 850g now, how much of it will there be in: (a)Six years time? (b)Ten years time? Example Question 8 The population of a small Island is expected to decrease steadily at a constant rate of 2% each year. If the population is currently 3600 what will it be in: (a)Twelve years time? (b)Twenty years time? Question 7 In an experiment a 500g mass of radio active copper decays at 4% per hour. How much copper is left after: (a)Three hours? (b)Eight hours? Question 8 The population of a small African country (2.73 million) is predicted to decline at a constant rate of 3% each year over the next 50 years. Calculate the expected population (nearest 1000) in: (a)Fifteen years time? (b)Thirty years time? Worksheet 4

25. Growth and Decay Graphs A culture of 800 bacteria increase at a constant rate of 14% per hour. (a)Complete the table above. (b)Draw the graph of the number of bacteria over time for the first five hours. (c)Use the graph to estimate the number of bacteria in the culture after 2½ hours. Time (hrs) 012345 Bacteria 800 0 1 2 3 4 5 Time (hours) Number of Bacteria Example Question 1

26. Growth and Decay Graphs Time (hrs) 012345 Mass (g) 600 0 1 2 3 4 5 Time (hours) Question 1 Mass of Metal (g) In an experiment a 600g mass of radio active metal decays at a constant rate of 20% per hour. (a)Complete the table above. (b)Draw the graph of mass over time for the first five hours. (c)Use the graph to estimate how long it takes for half the mass to decay.