Modelling with Functions (DP IB Maths: AI HL)

A fence of length is made to go around the perimeter of a rectangular paddock that borders a straight river. The cost of the fence along the river is per metre, while on the other three sides the cost is  per metre. The total cost of the fence is .

Calculate the maximum area of the paddock.

Using the value for the area from part (a), calculate

the total length of the fence.

Henry is about to start a 25-game rugby season for his school. Teams receive 4 points for a win and 2 points for a draw. No points are awarded for a loss.

Explain why  and   must also be conditions related to the problem.

Write down an equation for the number of points, P, a team receives from  wins and  draws.

After 10 games Henry’s team has lost 1 game and they have 32 points.

Find the number of wins and draws Henry’s team has had.

Antibiotics A and B are applied to a pure culture of bacteria. The number of bacteria present initially for both antibiotics is 6000. The number of bacteria present for antibiotic A, , can be modelled by the function

,

where  is the elapsed time, in hours, since the start of the experiment.

Find the value of .

The number of bacteria present for antibiotic A after two hours is 2160.

Find the value of . Give your answer as a fraction.

The number of bacteria present for antibiotic B after four hours is 1185. The number of bacteria present for antibiotic B can be modelled using a similar function to antibiotic A.

Write down the function .

Determine which antibiotic is more effective. Give a reason for your answer.

In 1967 a house was bought for $10 000.  In 2021 the same house was sold for $800 000.

The average annual growth percentage of the house from 1967 to 2021 is used to form the following model to estimate the value of the house.

where  is the time in years.

Find the value of  and  and write down the function .

In 2026 the house is bought for $1 300 000.

Calculate the percentage error between the actual value of the house and the estimated value approximated by the model.

A factory produces cardboard boxes in the shape of a cuboid, with a fixed height of 25 cm and a base of varying area. The area, , of each base can be modelled by the function

where  is the width of the base of the cardboard box in centimetres.

Cardboard box M has a width of 12 cm.

Find the volume of cardboard box M.

Find the possible dimensions of a cardboard box with volume of .

The downward speed, V, in metres per second, of a bird making a dive into the water to catch a fish can be modelled by the function

,

where , in seconds, is the time the bird is diving.

Write down the downward speed of the bird at 

Determine the equation of the horizontal asymptote for the graph of 

The bird’s downward speed when it reaches the surface of the ocean is .

Find the birds downward speed in kilometres per hour .

Find the time, in seconds, for which the bird was diving.

The temperature, T, of a cake, in degrees Celsius, , can be modelled by the function

where  is a constant and  is the time, in minutes, since the cake was taken out of the oven.

In the context of this model, state what the value of 18 represents.

The cake was  when it was taken out of the oven.

Find the value of .

Find the temperature of the cake half an hour after being taken out of the oven.

The cake is best eaten when its temperature is 75 to 95.

Calculate for how many minutes the cake’s temperature is within this range.

A hot winter soup has just been removed from the stove and is left outside to cool.  The soup’s temperature can be modelled by the function

where  is the time, in minutes, since the soup was removed from the stove.

The temperature outside is 12.

Write down the value of  and explain why it has this value.

Initially the temperature of the soup is .

Find the value of .

After two minutes the temperature of the soup is 60.

Find the value of .

After 15 minutes the soup is put into the fridge.

Calculate the temperature of the soup when it is put into the fridge.

The approximate quantity, , of a decaying radioactive substance can be modelled by the function

where  is the initial quantity and  is the elapsed time, in years.  is the half-life of the substance and is a measure of how many years it takes for the quantity of the radioactive substance to decrease by half.

Radioactive substance A has  and after 2 years the quantity remaining is 124.

Find . Give your answer to the nearest integer.

Calculate the approximate quantity of the radioactive substance after five years. Give your answer to the nearest integer.

After five years the quantity remaining of the radioactive substance is 25.

Calculate the percentage error between your approximate value found in part (b) and the exact value given above.

Deserts are known for having high daily temperature ranges. Erica monitors the temperature, in , on a particular day in a desert. The table below shows some of the information she recorded.

	Temperature	Time
Maximum
Minimum

Erica uses her observations to form the following model for the temperature, , during the day

,

where  is the elapsed time, in hours, since midnight.

Calculate the value of  when the maximum temperature occurs and fill in the time in the table above in am/pm format.

Find the values of

Erica goes exploring in the desert at 6:30 am and leaves once the temperature reaches .

John is a contracted carpenter who earns an annual salary of $55 000. Additionally, John sells innovative wooden art at a market. He goes to the market twice a month and he generally earns $450 from the market.

Estimate John’s total annual income.

John’s actual total annual income over the year is $68 000.

Calculate the percentage error between your answer in part (a) and John’s actual total annual income.

The following table shows different tax rates for residents of the country John lives in.

Income thresholds, in	Rate, %	Tax payable on this income
	0	0
	15	of amounts over
	25	of amounts over
	40	of amounts over

Calculate the amount of tax John has to pay. Give your answer to the nearest euro (€).

Phillip is kayaking to an island located 7 km from where he starts. The distance, D, in kilometres, that Phillip has travelled from his original position can be modelled by the exponential function

where  is the time in minutes since Phillip started kayaking.

State the value of  and explain what the value represents in the context of this question.

Find the value of .

After 15 minutes Phillip has travelled 5.5 km from his original position.

Find the value of .

When Phillip is less than 20 m away from the island he can stand up and walk his kayak ashore.

Calculate the time it takes Phillip before he stands up and walks ashore. Give your answer to the nearest minute.

The average temperature of a city, C, in degrees Celsius, fluctuates throughout a year and can be modelled by the function

where  is the elapsed time, in weeks, since the start of the year.

The average temperature of the city in week 4 is 27 degrees Celsius and in week 28 it is 12 degrees Celsius.

Find the value of , assuming there are 52 weeks in a year.

Write down two equations connecting   and  and find their values.

A restaurant stores food in a storage unit outside and when the average temperature of the city gets below 0 degrees Celsius they have to be careful about some things freezing.

Calculate how many weeks of the year that they have to be careful about the food freezing. Give your answer to the nearest integer.

Algae in a lake can grow exponentially until the lake is fully covered in algae.

Find the number of days it takes for a lake to be fully covered in algae when  of the lake is covered today and the covered area doubles once every five days.

Find the number of days it takes for a lake to be fully covered in algae when  of the lake is uncovered and the covered area increases by a factor of 10 within each 10-day period.

A fence of length , in metres, is made to form a rectangle around a house that borders a forest on one side. The fence does not run along the side next to the forest.  The cost of the fence is  per metre. The total cost of the fence is .

Calculate

A potato top pie is removed from the oven and is left to cool. The pie’s temperature, T, in , can be modelled by the function

where  is the time, in minutes, since the pie was removed from the oven.

The temperature of the kitchen is .

Write down the value of  and explain what it represents in the context of this question.

Initially the temperature of the pie is 

Find the value of .

After five minutes the temperature of the pie is .

Find the value of .

Bacteria in the pie can grow rapidly when its temperature is in the “danger zone” which is between  and . Food should never be left in the “danger zone” for more than 2 hours. Hence, the pie is put in the fridge after it has been in the “danger zone” for an hour and 20 minutes.

Calculate the total amount of time between the pie being removed from the oven and being put in the fridge. Give your answer to the nearest minute.

Matt throws a discus in a competition, and its flight can be modelled by the function

,

where  is the horizontal distance in metres from where the athlete threw the javelin and  is the height of the javelin above the ground in metres.

In the context of the model, explain the significance of the 1.8.

Sketch a graph of the model, labelling any intersections with the coordinate axes and the maximum point.

Matt has been training for this competition for 6 months and his improvement can be modelled by the function

where D is the distance of his personal best throw, in metres, and  is the time that has elapsed since he started training, in months and  and  are constants.

At the start of his training Matt’s personal best throw is 14 m. Matt’s throw in the competition was a new personal best.

Find the values of .

A tunnel is being constructed and its opening can be modelled by the quadratic function

where  is the height of the tunnel, in metres, and  is the width of the tunnel, in metres.

It is given that  and .

Find the values of  and .

The height required for a lane of traffic is 5 m and each lane requires a width of 2.8 m.

Find the number of lanes of traffic the tunnel can fit.

A company sells 55 cars per month for a sale price of $2000, whilst incurring costs for supplies, production and delivery of $890 per car. Reliable market research shows that for each increase (or decrease) of the sale price by $50 the company will sell 5 cars less (or more) and vice versa.

Find an expression for the total profit, P, in terms of the sale price, .

Find the values of  when  and explain their significance in the context of the question.

Calculate

A company sells  litres of water per month and their total monthly profit, , can be modelled by the function

where  is the sale price of each litre sold, in dollars, at and  is the linear function for the number of litres the company can sell per month at each given sale price.

In the context of the question, explain the significance of the 0.45.

It is given that  and .

Write down the function of , in the form , where  and  are constants.

Find the values of  when  and explain their significance in the context of the question.

Calculate