2.3 Modelling with Functions

The front view of the edge of a water tank is drawn on a set of axes below. The edge is modelled by 

Point  has coordinates point  has coordinates and point  has coordinates

Find the value of .

Find the value of .

Given that  unit represents  m, find the width of the water tank  when its height is 

The number of German words, , that Helen remembers after completing a German language course decreases exponentially over time when she does not practice her German. This decrease can be modelled by the function

Where  and  are positive constants and is the time in years since Helen completed the German language course.

Helen can remember  German words as soon as she completes the German language course.

Find the value of .

After  years Helen has not practiced her German and can only remember  German words.

Find the value of .

The number of German words Helen remembers never decreases below a certain number of words, .

Find the value of .

In a trial for a new drug, scientists found that the amount of the drug in the bloodstream decreased over time, according to the model

where is the amount of the drug in the bloodstream in mg per litre  and  is the time in hours.

Write down the amount of the drug in the bloodstream at

Calculate the amount of the drug in the bloodstream after four hours.

Calculate the time, in hours, for the amount of the drug in the bloodstream to decrease to .

The scientists found that some of the test subjects had an elevated heart rate for  minutes after ingesting the drug.

Find the amount of the drug in the bloodstream when the heart rates from the effected test subjects returned to normal.

The number of bacteria in a Petri dish is modelled by the function

where  is the number of bacteria and  is the time in hours.

Write down the number of bacteria in the Petri dish at 

Calculate the number of bacteria present after 

Calculate the time, in hours, for the number of bacteria to reach 

A remote-controlled sailboat’s velocity is dependent on the wind speed. The sailboat’s velocity is lower during very high and very low wind speeds.

The sailboat’s velocity can be modelled by the function

where V is the sailboat’s velocity, in , and  is the wind speed, in .

Find the sailboat’s velocity when the windspeed is .

Find the windspeed when the sailboat’s velocity is .

Show that .

Using your graphics display calculator find the maximum velocity of the sailboat and the windspeed required for this.

A Ferris wheel rotates at a constant speed, the height of a particular seat above the ground is modelled by the function

where  is the height of the seat above the ground, in metres, and  is the elapsed time, in seconds, since the start of the ride.

Write down

Calculate the number of seconds it takes for the Ferris wheel to do one full rotation.

The water depth, , in metres, at a port can be modelled by the function

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

The cycle of water depths repeats every hours. Find the value of 

A rectangular sheet of cardboard  cm by cm has square sides of cm cut from each corner. It is folded to make an open box as shown.

Show that the volume of the box can be modelled by the function

State the domain of .

Using your graphics display calculator find the maximum value of and the value of  which gives this volume 

Grace leaves a cup of hot tea to cool and measures its temperature every minute. Her results are shown in the table below.

Time,  (minutes)
Temperature,  (°C)

Write down the decrease in temperature of the tea

Assuming the pattern in the answers to part (a) continues, find the value of . Give your answer correct to  decimal places.

The function that models the change in temperature of the tea is , where represents the temperature the tea tends towards and is the initial temperature.

Write down two equations relating  and .

Find the value of  and .

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