Compound growth applications

7 A tool for putting fence posts into the ground is called a 'post-rammer'. The distances in millimetres that the post sinks into the ground on each impact of the post-rammer follow a geometric progression. The first three impacts cause the post to sink into the ground by \(50 \mathrm {~mm} , 40 \mathrm {~mm}\) and 32 mm respectively.

1. Verify that the 9th impact is the first in which the post sinks less than 10 mm into the ground.
2. Find, to the nearest millimetre, the total depth of the post in the ground after 20 impacts.
3. Find the greatest total depth in the ground which could theoretically be achieved.

3 Each year a company gives a grant to a charity. The amount given each year increases by \(5 \%\) of its value in the preceding year. The grant in 2001 was \(\\) 5000$. Find

1. the grant given in 2011,
2. the total amount of money given to the charity during the years 2001 to 2011 inclusive.

4

1. An arithmetic progression has a first term of 32, a 5th term of 22 and a last term of - 28 . Find the sum of all the terms in the progression.
2. Each year a school allocates a sum of money for the library. The amount allocated each year increases by \(2.5 \%\) of the amount allocated the previous year. In 2005 the school allocated \(\\) 2000$. Find the total amount allocated in the years 2005 to 2014 inclusive.

3 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.
2. Find the total amount of salt obtained in the first 12 weeks after the change.

6 A small trading company made a profit of \(\\) 250000\( in the year 2000. The company considered two different plans, plan \)A\( and plan \)B$, for increasing its profits. Under plan \(A\), the annual profit would increase each year by \(5 \%\) of its value in the preceding year. Find, for plan \(A\),

1. the profit for the year 2008,
2. the total profit for the 10 years 2000 to 2009 inclusive. Under plan \(B\), the annual profit would increase each year by a constant amount \(\\) D\(.
3. Find the value of \)D$ for which the total profit for the 10 years 2000 to 2009 inclusive would be the same for both plans.

8

1. A cyclist completes a long-distance charity event across Africa. The total distance is 3050 km . He starts the event on May 1st and cycles 200 km on that day. On each subsequent day he reduces the distance cycled by 5 km .
   1. How far will he travel on May 15th?
   2. On what date will he finish the event?
2. A geometric progression is such that the third term is 8 times the sixth term, and the sum of the first six terms is \(31 \frac { 1 } { 2 }\). Find
   1. the first term of the progression,
   2. the sum to infinity of the progression.

4 A runner who is training for a long-distance race plans to run increasing distances each day for 21 days. She will run \(x \mathrm {~km}\) on day 1 , and on each subsequent day she will increase the distance by \(10 \%\) of the previous day's distance. On day 21 she will run 20 km .

1. Find the distance she must run on day 1 in order to achieve this. Give your answer in km correct to 1 decimal place.
2. Find the total distance she runs over the 21 days.

12. A business is expected to have a yearly profit of \(\pounds 275000\) for the year 2016. The profit is expected to increase by \(10 \%\) per year, so that the expected yearly profits form a geometric sequence with common ratio 1.1

1. Show that the difference between the expected profit for the year 2020 and the expected profit for the year 2021 is \(\pounds 40300\) to the nearest hundred pounds.
2. Find the first year for which the expected yearly profit is more than one million pounds.
3. Find the total expected profits for the years 2016 to 2026 inclusive, giving your answer to the nearest hundred pounds.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The number of dormice and the number of voles on an island are being monitored.
Initially there are 2000 dormice on the island.
A model predicts that the number of dormice will increase by \(3 \%\) each year, so that the numbers of dormice on the island at the end of each year form a geometric sequence.

1. Find, according to the model, the number of dormice on the island 6 years after monitoring began. Give your answer to 3 significant figures. The number of voles on the island is being monitored over the same period of time.
   Given that
   • 4 years after monitoring began there were 3690 voles on the island
   • 7 years after monitoring began there were 3470 voles on the island
   • the number of voles on the island at the end of each year is modelled as a geometric sequence
   • find the equation of this model in the form
   $$N = a b ^ { t }$$ where \(N\) is the number of voles, \(t\) years after monitoring began and \(a\) and \(b\) are constants. Give the value of \(a\) and the value of \(b\) to 2 significant figures. When \(t = T\), the number of dormice on the island is equal to the number of voles on the island.
2. Find, according to the models, the value of \(T\), giving your answer to one decimal place.

6. A car was purchased for \(\pounds 18000\) on 1 st January. On 1st January each following year, the value of the car is \(80 \%\) of its value on 1st January in the previous year.

1. Show that the value of the car exactly 3 years after it was purchased is \(\pounds 9216\). The value of the car falls below \(\pounds 1000\) for the first time \(n\) years after it was purchased.
2. Find the value of \(n\). An insurance company has a scheme to cover the maintenance of the car. The cost is \(\pounds 200\) for the first year, and for every following year the cost increases by \(12 \%\) so that for the 3rd year the cost of the scheme is \(\pounds 250.88\)
3. Find the cost of the scheme for the 5th year, giving your answer to the nearest penny.
4. Find the total cost of the insurance scheme for the first 15 years.
   \section*{LU}

3. A company predicts a yearly profit of \(\pounds 120000\) in the year 2013 . The company predicts that the yearly profit will rise each year by \(5 \%\). The predicted yearly profit forms a geometric sequence with common ratio 1.05

1. Show that the predicted profit in the year 2016 is \(\pounds 138915\)
2. Find the first year in which the yearly predicted profit exceeds \(\pounds 200000\)
3. Find the total predicted profit for the years 2013 to 2023 inclusive, giving your answer to the nearest pound.

6. At the beginning of the year 2000 a company bought a new machine for \(\pounds 15000\). Each year the value of the machine decreases by \(20 \%\) of its value at the start of the year.

1. Show that at the start of the year 2002, the value of the machine was \(\pounds 9600\). When the value of the machine falls below \(\pounds 500\), the company will replace it.
2. Find the year in which the machine will be replaced. To plan for a replacement machine, the company pays \(\pounds 1000\) at the start of each year into a savings account. The account pays interest at a fixed rate of \(5 \%\) per annum. The first payment was made when the machine was first bought and the last payment will be made at the start of the year in which the machine is replaced.
3. Using your answer to part (b), find how much the savings account will be worth immediately after the payment at the start of the year in which the machine is replaced.

8 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 100000 barrels.

1. Calculate the amount pumped in 2004. 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.
2. Calculate in which year the amount pumped will fall below 5000 barrels.
3. Calculate the total amount of oil pumped from the well from the year 2001 up to and including the final year of production.

11 When Fred joined a computer firm his salary was \(\pounds 28000\) per annum. In each subsequent year he received an annual increase of \(12 \%\) of his previous year's salary.

1. State Fred's salary for each of his first 3 years with the company. State also the common ratio of the geometric sequence formed by his salaries.
2. How much did Fred earn in the tenth year?
3. Show that the total amount Fred earned over the ten years was between \(\pounds 400000\) and £500000.
4. When Fred joined the computer firm, his brother Archie set up a plumbing business. He earned \(\pounds 35000\) in his first year and each year earned \(\pounds d\) more than in the previous year. At the end of ten years, he had earned exactly the same total amount as Fred. Calculate the value of \(d\).

7. \includegraphics[max width=\textwidth, alt={}, center]{e4afa57d-5be3-42a6-ab35-39b0fdcc1681-2_364_666_1338_568} The diagram shows part of a design being produced by a computer program.
The program draws a series of circles with each one touching the previous one and such that their centres lie on a horizontal straight line. The radii of the circles form a geometric sequence with first term 1 mm and second term 1.5 mm . The width of the design is \(w\) as shown.

1. Find the radius of the fourth circle to be drawn.
2. Show that when eight circles have been drawn, \(w = 98.5 \mathrm {~mm}\) to 3 significant figures.
3. Find the total area of the design in square centimetres when ten circles have been drawn.

4 There is a flowerhead at the end of each stem of an oleander plant. The next year, each flowerhead is replaced by three stems and flowerheads, as shown in Fig. 11. \begin{figure}[h] \end{figure}

1. How many flowerheads are there in year 5 ?
2. How many flowerheads are there in year \(n\) ?
3. As shown in Fig. 11, the total number of stems in year 2 is 4, (that is, 1 old one and 3 new ones). Similarly, the total number of stems in year 3 is 13 , (that is, \(1 + 3 + 9\) ). Show that the total number of stems in year \(n\) is given by \(\frac { 3 ^ { n } - 1 } { 2 }\).
4. Kitty's oleander has a total of 364 stems. Find
   (A) its age,
   (B) how many flowerheads it has.
5. Abdul's oleander has over 900 flowerheads. Show that its age, \(y\) years, satisfies the inequality \(y > \frac { \log _ { 10 } 900 } { \log _ { 10 } 3 } + 1\).
   Find the smallest integer value of \(y\) for which this is true.

11 Jill has 3 daughters and no sons. They are generation 1 of Jill's descendants.
Each of her daughters has 3 daughters and no sons. Jill's 9 granddaughters are generation 2 of her descendants. Each of her granddaughters has 3 daughters and no sons; they are descendant generation 3. Jill decides to investigate what would happen if this pattern continues, with each descendant having 3 daughters and no sons.

1. How many of Jill's descendants would there be in generation 8 ?
2. How many of Jill's descendants would there be altogether in the first 15 generations?
3. After \(n\) generations, Jill would have over a million descendants altogether. Show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 2000003 } { \log _ { 10 } 3 } - 1 .$$ Hence find the least possible value of \(n\).
4. How many fewer descendants would Jill have altogether in 15 generations if instead of having 3 daughters, she and each subsequent descendant has 2 daughters? \section*{END OF QUESTION PAPER}

7 Chris runs half marathons, and is following a training programme to improve his times. His time for his first half marathon is 150 minutes. His time for his second half marathon is 147 minutes. Chris believes that his times can be modelled by a geometric progression.

1. Chris sets himself a target of completing a half marathon in less than 120 minutes. Show that this model predicts that Chris will achieve his target on his thirteenth half marathon.
2. After twelve months Chris has spent a total of 2974 minutes, to the nearest minute, running half marathons. Use this model to find how many half marathons he has run.
3. Give two reasons why this model may not be appropriate when predicting the time for a half marathon.

1. In this question you should show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} A company made a profit of \(\pounds 20000\) in its first year of trading, Year 1
A model for future trading predicts that the yearly profit will increase by \(8 \%\) each year, so that the yearly profits will form a geometric sequence. According to the model,

1. show that the profit for Year 3 will be \(\pounds 23328\)
2. find the first year when the yearly profit will exceed £65000
3. find the total profit for the first 20 years of trading, giving your answer to the nearest £1000

13 The population of Melchester is 185207. During a nationwide flu epidemic the number of new cases in Melchester are recorded each day. The results from the first three days are shown in Fig. 13. \begin{table}[h] \end{table} A doctor notices that the numbers of new cases on successive days are in geometric progression.

1. Find the common ratio for this geometric progression. The doctor uses this geometric progression to model the number of new cases of flu in Melchester.
2. According to the model, how many new cases will there be on day 5?
3. Find a formula for the total number of cases from day 1 to day \(n\) inclusive according to this model, simplifying your answer.
4. Determine the maximum number of days for which the model could be viable in Melchester.
5. State, with a reason, whether it is likely that the model will be viable for the number of days found in part (d).

7. A student completes a mathematics course and begins to work through past exam papers. He completes the first paper in 2 hours and the second in 1 hour 54 minutes. Assuming that the times he takes to complete successive papers form a geometric sequence,

1. find, to the nearest minute, how long he will take to complete the fifth paper,
2. show that the total time he takes to complete the first eight papers is approximately 13 hours 28 minutes,
3. find the least number of papers he must work through if he is to complete a paper in less than one hour.

1. A geometric series has first term \(a\) and common ratio \(r\). Prove that the sum of the first \(n\) terms of the series is $$\frac{a(1-r^n)}{1-r}.$$ [4]

Mr King will be paid a salary of £35 000 in the year 2005. Mr King's contract promises a 4% increase in salary every year, the first increase being given in 2006, so that his annual salaries form a geometric sequence.

1. Find, to the nearest £100, Mr King's salary in the year 2008. [2]

Mr King will receive a salary each year from 2005 until he retires at the end of 2024.

1. Find, to the nearest £1000, the total amount of salary he will receive in the period from 2005 until he retires at the end of 2024. [4]

A building has a leaking roof and, while it is raining, water drips into a 12 litre bucket. When the rain stops, the bucket is one third full. Water continues to drip into the bucket from a puddle on the roof. In the first minute after the rain stops, 30 millilitres of water drips into the bucket. In each subsequent minute, the amount of water that drips into the bucket reduces by 2%. During the \(n\)th minute after the rain stops, the volume of water that drips into the bucket is \(W_n\) millilitres.

1. Find \(W_2\) [1 mark]
2. Explain why $$W_n = A \times 0.98^{n-1}$$ and state the value of \(A\). [2 marks]
3. Find the increase in the water in the bucket 15 minutes after the rain stops. Give your answer to the nearest millilitre. [2 marks]
4. Assuming it does not start to rain again, find the maximum amount of water in the bucket. [3 marks]
5. After several hours the water has stopped dripping. Give two reasons why the amount of water in the bucket is not as much as the answer found in part (d). [2 marks]

A tree is 80 cm in height when it is planted. In the first year, the tree grows in height by 32 cm. In each subsequent year, the tree grows in height by 90% of the growth of the previous year.

1. Find the height of the tree 10 years after it was planted. [4]
2. Determine the maximum height of the tree. [2]

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]