1.04k Modelling with sequences: compound interest, growth/decay

A company producing salt from sea water changed to a new process. The amount of salt obtained each week increased by 2% of the amount obtained in the preceding week. It is given that in the first week after the change the company obtained 8000 kg of salt.

1. Find the amount of salt obtained in the 12th week after the change. [3]
2. Find the total amount of salt obtained in the first 12 weeks after the change. [2]

The amounts of oil pumped from an oil well in each of the years 2001 to 2004 formed a geometric progression with common ratio 0.9. The amount pumped in 2001 was 100 000 barrels.

1. Calculate the amount pumped in 2004. [2]

It is assumed that the amounts of oil pumped in future years will continue to follow the same geometric progression. Production from the well will stop at the end of the first year in which the amount pumped is less than 5000 barrels.

1. Calculate in which year the amount pumped will fall below 5000 barrels. [4]
2. Calculate the total amount of oil pumped from the well from the year 2001 up to and including the final year of production. [3]

Robert wants to deposit \(£P\) into a savings account. He has a choice of two accounts. • Account \(A\) offers an annual compound interest rate of \(1\%\). • Account \(B\) offers an interest rate of \(5\%\) for the first year and an annual compound interest rate of \(0.6\%\) for each subsequent year. After \(n\) years, account \(A\) is more profitable than account \(B\). Find the smallest value of \(n\). [5]

A ball is released from rest from a height of 5 m and bounces repeatedly on horizontal ground. After hitting the ground for the first time, the ball rises to a maximum height of 3 m. In a model for the motion of the ball • the maximum height after each bounce is 60% of the previous maximum height • the motion takes place in a vertical line

1. Using the model
   1. show that the maximum height after the 3rd bounce is 1.08 m,
   2. find the total distance the ball travels from release to when the ball hits the ground for the 5th time.
   [3] According to the model, after the ball is released, there is a limit, \(D\) metres, to the total distance the ball will travel.
2. Find the value of \(D\) [2] With reference to the model,
3. give a reason why, in reality, the ball will not travel \(D\) metres in total. [1]

Liquid is kept in containers, which due to evaporation and ongoing chemical reactions, at the end of each month the volume of the liquid in these containers reduces by 10% compared with the volume at the start of the same month. One such container is filled up with 250 litres of liquid.

1. Show that the volume of the liquid in the container at the end of the second month is 202.5 litres. [1]
2. Find the volume of the liquid in the container at the end of the twelfth month. [2]

At the start of each month a new container is filled up with 250 litres of liquid, so that at the end of twelve months there are 12 containers with liquid.

1. Use an algebraic method to calculate the total amount of liquid in the 12 containers at the end of 12 months. [5]