Total over time period

12. Karen is going to raise money for a charity. She aims to cycle a total distance of 1000 km over a number of days.
On day one she cycles 25 km .
She increases the distance that she cycles each day by \(10 \%\) of the distance cycled on the previous day, until she reaches the total distance of 1000 km . She reaches the total distance of 1000 km on day \(N\), where \(N\) is a positive integer.

1. Find the value of \(N\). On day one, 50 people donated money to the charity. Each day, 20 more people donated to the charity than did so on the previous day, so that 70 people donated money on day two, 90 people donated money on day three, and so on.
2. Find the number of people who donated to the charity on day fifteen. Each day, the donation given by each person was \(\pounds 5\)
3. Find the total amount of money donated by the end of day fifteen.

11. Wheat is to be grown on a farm. A model predicts that the mass of wheat harvested on the farm will increase by \(1.5 \%\) per year, so that the mass of wheat harvested each year forms a geometric sequence. Given that the mass of wheat harvested during year one is 6000 tonnes,

1. show that, according to the model, the mass of wheat harvested on the farm during year 4 will be approximately 6274 tonnes. During year \(N\), according to the model, there is predicted to be more than 8000 tonnes of wheat harvested on the farm.
2. Find the smallest possible value of \(N\). It costs \(\pounds 5\) per tonne to harvest the wheat.
3. Assuming the model, find the total amount that it would cost to harvest the wheat from year one to year 10 inclusive. Give your answer to the nearest \(\pounds 1000\).

2. The adult population of a town at the start of 2019 is 25000 A model predicts that the adult population will increase by \(2 \%\) each year, so that the number of adults in the population at the start of each year following 2019 will form a geometric sequence.

1. Find, according to the model, the adult population of the town at the start of 2032 It is also modelled that every member of the adult population gives \(\pounds 5\) to local charity at the start of each year.
2. Find, according to these models, the total amount of money that would be given to local charity by the adult population of the town from 2019 to 2032 inclusive. Give your answer to the nearest \(\pounds 1000\)

9. The adult population of a town is 25000 at the end of Year 1. A model predicts that the adult population of the town will increase by \(3 \%\) each year, forming a geometric sequence.

1. Show that the predicted adult population at the end of Year 2 is 25750.
2. Write down the common ratio of the geometric sequence. The model predicts that Year \(N\) will be the first year in which the adult population of the town exceeds 40000.
3. Show that $$( N - 1 ) \log 1.03 > \log 1.6$$
4. Find the value of \(N\). At the end of each year, each member of the adult population of the town will give \(\pounds 1\) to a charity fund. Assuming the population model,
5. find the total amount that will be given to the charity fund for the 10 years from the end of Year 1 to the end of Year 10, giving your answer to the nearest \(\pounds 1000\).

1. A competitor is running a 20 kilometre race.

She runs each of the first 4 kilometres at a steady pace of 6 minutes per kilometre. After the first 4 kilometres, she begins to slow down. In order to estimate her finishing time, the time that she will take to complete each subsequent kilometre is modelled to be \(5 \%\) greater than the time that she took to complete the previous kilometre. Using the model,

1. show that her time to run the first 6 kilometres is estimated to be 36 minutes 55 seconds,
2. show that her estimated time, in minutes, to run the \(r\) th kilometre, for \(5 \leqslant r \leqslant 20\), is $$6 \times 1.05 ^ { r - 4 }$$
3. estimate the total time, in minutes and seconds, that she will take to complete the race.

1. There were 2100 tonnes of wheat harvested on a farm during 2017.

The mass of wheat harvested during each subsequent year is expected to increase by \(1.2 \%\) per year.

1. Find the total mass of wheat expected to be harvested from 2017 to 2030 inclusive, giving your answer to 3 significant figures. Each year it costs
   • £5.15 per tonne to harvest the first 2000 tonnes of wheat
   • £6.45 per tonne to harvest wheat in excess of 2000 tonnes
   • Use this information to find the expected cost of harvesting the wheat from 2017 to 2030 inclusive. Give your answer to the nearest \(\pounds 1000\)

8. Amy plans to join a savings scheme in which she will pay in \(\pounds 500\) at the start of each year. One scheme that she is considering pays 6\% interest on the amount in the account at the end of each year. For this scheme,

1. find the amount of interest paid into the account at the end of the second year,
2. show that after interest is paid at the end of the eighth year, the amount in the account will be \(\pounds 5246\) to the nearest pound. Another scheme that she is considering pays \(0.5 \%\) interest on the amount in the account at the end of each month.
3. Find, to the nearest pound, how much more or less will be in the account at the end of the eighth year under this scheme.

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]

Records are kept of the number of copies of a certain book that are sold each week. In the first week after publication 3000 copies were sold, and in the second week 2400 copies were sold. The publisher forecasts future sales by assuming that the number of copies sold each week will form a geometric progression with first two terms 3000 and 2400. Calculate the publisher's forecasts for

1. the number of copies that will be sold in the 20th week after publication, [3]
2. the total number of copies sold during the first 20 weeks after publication, [2]
3. the total number of copies that will ever be sold. [2]

On the first day of each month, Kate pays £50 into a savings account. Interest is paid on the total amount in the account on the last day of each month. The interest rate is 0.2% At the end of the \(n\)th month, the total amount of money in Kate's savings account is £\(T_n\) Kate correctly calculates \(T_1\) and \(T_2\) as shown below: \(T_1 = 50 \times 1.002 = 50.10\) \(T_2 = (T_1 + 50) \times 1.002\) \(= ((50 \times 1.002) + 50) \times 1.002\) \(= 50 \times 1.002^2 + 50 \times 1.002\) \(\approx 100.30\)

1. Show that \(T_3\) is given by $$T_3 = 50 \times 1.002^3 + 50 \times 1.002^2 + 50 \times 1.002$$ [1 mark]
2. Kate uses her method to correctly calculate how much money she can expect to have in her savings account at the end of 10 years.
   1. Find the amount of money Kate expects to have in her savings account at the end of 10 years. [3 marks]
   2. The amount of money in Kate's savings account at the end of 10 years may not be the amount she has correctly calculated. Explain why. [1 mark]

Aled decides to invest £1000 in a savings scheme on the first day of each year. The scheme pays 8% compound interest per annum, and interest is added on the last day of each year. The amount of savings, in pounds, at the end of the third year is given by $$1000 \times 1 \cdot 08 + 1000 \times 1 \cdot 08^2 + 1000 \times 1 \cdot 08^3$$ Calculate, to the nearest pound, the amount of savings at the end of thirty years. [5]

A company which makes batteries for electric cars has a 10-year plan for growth. • In year 1 the company will make 2600 batteries • In year 10 the company aims to make 12000 batteries In order to calculate the number of batteries it will need to make each year, from year 2 to year 9, the company considers the following model: *the number of batteries made will increase by the same percentage each year* Showing detailed reasoning, calculate the total number of batteries made from year 1 to year 10. [3]

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]