Recurrence Relations

A population of $\displaystyle 7400$ decreases by $\displaystyle 12\%$ each year.
At the end of each year $\displaystyle 260$ individuals join the population.
A spreadsheet models the population year by year, with the starting value in cell B2.
Write down the formula that should be entered in cell B3 so that it can be filled down the column.

A savings account contains $\displaystyle \pounds 2000$.
Each year $\displaystyle 4\%$ interest is added and then $\displaystyle \pounds 150$ is withdrawn.
A spreadsheet models the balance year by year, with the starting value in cell B2.
Write down the formula that should be entered in cell B3 so that it can be filled down the column.

A car park contains $\displaystyle 400$ cars at $\displaystyle 8$ am.
During each hour, $\displaystyle 30\%$ of the cars leave and $\displaystyle 60$ cars arrive.
Calculate the number of cars in the car park at $\displaystyle 9$ am and at $\displaystyle 10$ am.

A lake contains $\displaystyle 12000$ fish.
Each year $\displaystyle 15\%$ of the fish are caught and $\displaystyle 1500$ young fish are added.
Calculate the number of fish after $\displaystyle 3$ years.
Give your answer to the nearest whole fish.

A tank holds $\displaystyle 800$ litres of liquid.
Each day $\displaystyle 20\%$ of the liquid evaporates and $\displaystyle 100$ litres are added.
A spreadsheet models the volume day by day, with the starting value in cell B2.
(a) Write down the formula that should be entered in cell B3 so that it can be filled down the column.
(b) Describe how to use the spreadsheet to find what happens to the volume in the long term, and state the outcome.

A spreadsheet models the mass of a chemical in a tank.
Each hour, $\displaystyle 40\%$ of the chemical is removed and $\displaystyle 200$ grams are added.
The tank starts with $\displaystyle 1000$ grams.
Calculate the mass of the chemical after each of the next three hours.

A forest contains $\displaystyle 40000$ trees.
Each year $\displaystyle 8\%$ of the trees are felled and $\displaystyle 2000$ saplings are planted.
(a) Calculate the number of trees after $\displaystyle 2$ years.
(b) Describe how a spreadsheet could be used to find the number of trees in the forest in the long term.

An orchard is sprayed weekly with a pesticide that destroys $\displaystyle 60\%$ of the pests.
Between sprayings $\displaystyle 500$ new pests arrive.
There are $\displaystyle 3000$ pests before the first spraying.
Calculate the number of pests after $\displaystyle 2$ weeks.

A spreadsheet models a population that falls by $\displaystyle 14\%$ each year, with $\displaystyle 120$ new members added each year.
Cell B2 holds the starting value.
Cell D1 holds the multiplier $\displaystyle 0.86$ and cell D2 holds $\displaystyle 120$.
Write down the formula that should be entered in cell B3 so that it can be filled down the column.

You must refer to the spreadsheet file 'Q5 School Roll.xlsx' when answering this question.

A school is planning a new building as it is approaching its maximum capacity.

The school roll in August 2021 was 650 pupils.

Approximately 18% of pupils leave by the end of each school year. 140 new S1 pupils join the roll in August each year.

(a)(i) Complete the 'School Roll' worksheet to predict the school roll in August 2031.

(a)(ii) Comment on the precision of this prediction.

(b) Comment on the relationship between time and the predicted school roll up to August 2031.

The school moves forward with plans for a new building. This will increase the capacity of the school to 800 pupils.

(c)(i) Extend the table in your worksheet to construct a graph to show what is predicted to happen to the school roll in the long term. You must consider what happens to the school roll beyond August 2031.

(c)(ii) Use your graph to determine whether the new capacity is suitable.

You must refer to the spreadsheet file 'Q8 Warehouse.xlsx' when answering this question.
You must complete parts (a) and (b) (i) using the spreadsheet file.
Parts (b) (ii), (c) and (d) must be completed in the answer space provided.

A warehouse company currently has 1750 units of stock. They deliver 20% of their current stock each week, and receive 300 units of new stock per week.

(a) Complete the 'Warehouse Stock' worksheet to identify (in cell C10) the predicted number of units of stock in the warehouse at the end of week 26.

(b)(i) Extend the table in your worksheet to construct a graph to show the units of stock for 52 weeks.

(b)(ii) Using the graph constructed in (b) (i), state which type of mathematical model best describes the units of stock in the warehouse.

After 52 weeks, the warehouse company plans to move to a smaller building with space for 1400 units of stock.

(c) Comment on whether the smaller building will have enough space for the stock. Justify your answer.

(d) State one reason why this mathematical model may not be realistic.

You must refer to the information on 'Dracaena plants and atmospheric carbon dioxide levels' given in the pre-release material when answering this question.

You must also refer to the spreadsheet file 'Q4 Dracaena Plant.xlsx' when answering this question. You must complete parts (a) and (b) using the spreadsheet file.

A research scientist is studying the effect of Dracaena plants to improve indoor air quality in a kitchen showroom.

They estimate that:

• large Dracaena plants reduce the concentration of carbon dioxide $\displaystyle (CO_2)$ in the showroom by 13% each day
• each evening a heating system is left running. This adds enough $\displaystyle CO_2$ to increase the concentration in the showroom by 180 ppm.

At the start of the study the concentration of $\displaystyle CO_2$ in the showroom was 2000 ppm.

(a) Complete the 'Dracaena plant study' worksheet to estimate the concentration of $\displaystyle CO_2$ at the end of 30 days.

The research scientist investigates the long-term concentration of $\displaystyle CO_2$ in the showroom.

(b) Extend the table in your worksheet to the end of 60 days. Construct a graph to show what will happen to the concentration of $\displaystyle CO_2$ in the showroom. Your graph must include an appropriate title and axes labels.

(c) Explain whether the large Dracaena plants are effective at obtaining very good indoor air quality in the showroom.