This case study explores exponential growth in two different examples. Rice According to folklore the man who invented chess was granted the right to name his reward by the King who was extremely impressed by the game. The clever man was also a mathematician and asked for this,’one grain of rice on the first square on the chess board, two grains on the second square, four on the third square and so on until the final square.’ The King apparently was unaware of just what this meant and asked his treasurer to count out the grains for him. When this was not completed a week later the King wanted to know why. Investigate how many grains of rice would be needed for the different squares of the chess board. You might just want to start with the first 16. Think about how many you might expect on the 32nd. Find a formula of the form R = ab”-1 to calculate the number of grains of rice (R) in square n. Convert this to be of the form R= Ae(n-1). Use this to calculate how many are on the 320d square and then the 64 h. Can you calculate how many grains there would be in total? Use this to write an explanation of why it was not possible to pay the inventor of chess the reward he asked for. You might want to research what this quantity of rice might look like. How much would it weigh? How much room would it take up? Is there even this much rice grown in the world? Savings Savings accumulate in a similar way; they do not double each year, but they do increase at a given rate each year depending on current interest rates. Interest rates vary between bank accounts and can vary between years. For this investigation we shall assume that the interest rates stay the same over the number of years we are investigating. Currently it is possible to save £15.000 a year in an ISA (a savings account where you do not pay tax on the interest). The amount of interest you can get often depends on whether or not you want access to your money. If you want easy access this tends to lower the amount of interest you can get. For the following accounts explore how much money you would have after 1, 3, 5 and 10 years. Can you find an exponential formula for each one? Easy access account with 1.5% interest rate 40-day notice account with 2% interest rate Fixed term account with 2.25% interest rate of just what this meant and asked his treasurer to count out the grains for him. When this was not completed a week later the King wanted to know why. Investigate how many grains of rice would be needed for the different squares of the chess board. You might just want to start with the first 16. Think about how many you might expect on the 32nd. Find a formula of the form R = abn-1 to calculate the number of grains of rice (R) in square n. Convert this to be of the form R= Ae8(n-1). Use this to calculate how many are on the 320 square and then the 64″. Can you calculate how many grains there would be in total? Use this to write an explanation of why it was not possible to pay the inventor of chess the reward he asked for. You might want to research what this quantity of rice might look like. How much would it weigh? How much room would it take up? Is there even this much rice grown in the world? Savings Savings accumulate in a similar way; they do not double each year, but they do increase at a given rate each year depending on current interest rates. Interest rates vary between bank accounts and can vary between years. For this investigation we shall assume that the interest rates stay the same over the number of years we are investigating. Currently it is possible to save £15,000 a year in an ISA (a savings account where you do not pay tax on the interest). The amount of interest you can get often depends on whether or not you want access to your money. If you want easy access this tends to lower the amount of interest you can get. For the following accounts explore how much money you would have after 1, 3, 5 and 10 years. Can you find an exponential formula for each one? Easy access account with 1.5% interest rate 40-day notice account with 2% interest rate Fixed term account with 2.25% interest rate Compare the differences between the accounts and explain the effect that wanting access to your money has on the accounts. Why do you think this is the case?
