Modelling with functions

A scientist is monitoring the flight of sea birds after they leave their nests on a cliff. The height above the sea, h metres, of one of the birds is modelled by the equation

where t seconds is the time after the bird leaves its nest and A, B and T are positive constants. Given that

• the bird was 17.6 m above the sea exactly 4 seconds after leaving its nest

• the bird was 11.9 m above the sea exactly 9 seconds after leaving its nest

find a complete equation for the model. Find, according to the model,

the height of the bird’s nest above the sea,

the limitation on the value of T.

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A new type of car is released for sale. The total number of this type of car sold, N, in a particular region, t months after the cars were released for sale, is modelled by the equation

Use the equation of the model to answer parts (a), (b), (c) and (d).

Find the total number of cars sold in the first 3 months. Given that N = 3000 when t = T

find the value of T giving the answer to 2 decimal places.

Find the rate of increase in the total number of cars sold when t = 3, giving the answer to 3 significant figures. After a marketing campaign, the total number of cars sold is expected to rise and have an upper limit of 6500

Using this information, suggest one refinement to the model.

A company is trying to determine the most profitable selling price for a new toy. Given that

• if the selling price of each toy is £30, the company expects to sell 1500 toys in one year

• if the selling price of each toy is £50, the company expects to sell 300 toys in one year

Using a linear model, with y being the expected number of toys sold in one year and x pounds being the selling price of the toy,

find an equation for y in terms of x. Given that

• the cost of making each toy is £10

• the company has additional costs of £8 000 per year

show that, according to the model, the yearly profit, P, in thousands of pounds, is given by

Use the model given in part (b) to answer parts (c), (d) and (e). Given that the company wishes to make a profit on sales of the toy,

find the range of possible selling prices of the toy.

Hence, or otherwise, deduce the selling price of the toy that maximises the profit. In one particular year, the company sold the toy for £35 and made £21 750 profit.

Use this information to evaluate the suitability of the model.

The graph in Figure 3 shows the path of a small ball.

The ball travels in a vertical plane above horizontal ground.

The ball is thrown from the point represented by A and caught at the point represented by B.

The height, H metres, of the ball above the ground has been plotted against the horizontal distance, x metres, measured from the point where the ball was thrown.

With respect to a fixed origin O, the point A has coordinates (0, 2) and the point B has coordinates (20, 0.8), as shown in Figure 3.

The ball reaches its maximum height when x = 9

A quadratic function, linking H with x, is used to model the path of the ball.

Find H in terms of x.

Give one limitation of the model.

Chandra is standing directly under the path of the ball at a point 16 m horizontally from O.

Chandra can catch the ball if the ball is less than 2.5 m above the ground.

Use the model to determine if Chandra can catch the ball.

The value, V pounds, of a mobile phone, t months after it was bought, is modelled by

where a and b are constants.

Figure 2 shows the linear relationship between log₁₀V and t. The line passes through the points (0, 3) and (10, 2.79) as shown.

Using these points,

find the initial value of the phone,

find a complete equation for V in terms of t, giving the exact value of a and giving the value of b to 3 significant figures.

Exactly 2 years after it was bought, the value of the phone was £320

Use this information to evaluate the reliability of the model.

On a roller coaster ride, passengers travel in carriages around a track.

On the ride, carriages complete multiple circuits of the track such that

• the maximum vertical height of a carriage above the ground is 60 m

• a carriage starts a circuit at a vertical height of 2 m above the ground

• the ground is horizontal

The vertical height, Hm, of a carriage above the ground, t seconds after the carriage starts the first circuit, is modelled by the equation

where a and b are positive constants.

Find a complete equation for the model.

Use the model to determine the height of the carriage above the ground whent = 40

In an alternative model, the vertical height, Hm, of a carriage above the ground, t seconds after the carriage starts the first circuit, is given by

where α and β are constants.

Find a complete equation for the alternative model.

Given that the carriage moves continuously for 2 minutes,

give a reason why the alternative model would be more appropriate.

Figure 3 is a graph of the trajectory of a golf ball after the ball has been hit until it first hits the ground. The vertical height, H metres, of the ball above the ground has been plotted against the horizontal distance travelled, x metres, measured from where the ball was hit. The ball is modelled as a particle travelling in a vertical plane above horizontal ground. Given that the ball

• is hit from a point on the top of a platform of vertical height 3 m above the ground

• reaches its maximum vertical height after travelling a horizontal distance of 90 m

• is at a vertical height of 27 m above the ground after travelling a horizontal distance of 120 m

Given also that H is modelled as a quadratic function in x

find H in terms of x

Hence find, according to the model,

the maximum vertical height of the ball above the ground,

the horizontal distance travelled by the ball, from when it was hit to when it first hits the ground, giving your answer to the nearest metre.

The possible effects of wind or air resistance are two limitations of the model. Give one other limitation of this model.

Figure 3 is a graph of the trajectory of a golf ball after the ball has been hit until it first hits the ground.

The vertical height, H metres, of the ball above the ground has been plotted against the horizontal distance travelled, x metres, measured from where the ball was hit.

The ball is modelled as a particle travelling in a vertical plane above horizontal ground.

Given that the ball

• is hit from a point on the top of a platform of vertical height 3 m above the ground

• reaches its maximum vertical height after travelling a horizontal distance of 90 m

• is at a vertical height of 27 m above the ground after travelling a horizontal distance of 120 m

Given also that H is modelled as a quadratic function in x

find H in terms of x

Hence find, according to the model,

the maximum vertical height of the ball above the ground,

the horizontal distance travelled by the ball, from when it was hit to when it first hits the ground, giving your answer to the nearest metre.

The possible effects of wind or air resistance are two limitations of the model. Give one other limitation of this model.

Express 2cosθ – sinθ in the form Rcos (θ + α), where R > 0 and 0 α Give the exact value of R and the value of α in radians to 3 decimal places.

Figure 6 shows the cross-section of a water wheel. The wheel is free to rotate about a fixed axis through the point C. The point P is at the end of one of the paddles of the wheel, as shown in Figure 6. The water level is assumed to be horizontal and of constant height. The vertical height, H metres, of P above the water level is modelled by the equation H = 3 + 4 cos (0.5 t) – 2 sin (0.5 t) where t is the time in seconds after the wheel starts rotating. Using the model, find

the maximum height of P above the water level,

the value of t when this maximum height first occurs, giving your answer to one decimal place. In a single revolution of the wheel, P is below the water level for a total of T seconds. According to the model,

find the value of T giving your answer to 3 significant figures. (Solutions based entirely on calculator technology are not acceptable.)

In reality, the water level may not be of constant height.

Explain how the equation of the model should be refined to take this into account.

Express 2cosθ – sinθ in the form Rcos (θ + α), where R > 0 and 0 α Give the exact value of R and the value of α in radians to 3 decimal places.

Figure 6 shows the cross-section of a water wheel.

The wheel is free to rotate about a fixed axis through the point C.

The point P is at the end of one of the paddles of the wheel, as shown in Figure 6.

The water level is assumed to be horizontal and of constant height.

The vertical height, H metres, of P above the water level is modelled by the equation

H = 3 + 4 cos (0.5 t) – 2 sin (0.5 t)

where t is the time in seconds after the wheel starts rotating.

Using the model, find

the maximum height of P above the water level,

the value of t when this maximum height first occurs, giving your answer to one decimal place.

In a single revolution of the wheel, P is below the water level for a total of T seconds.

According to the model,

find the value of T giving your answer to 3 significant figures.

(Solutions based entirely on calculator technology are not acceptable.)

In reality, the water level may not be of constant height.

Explain how the equation of the model should be refined to take this into account.

f(x) = 10e^–0.25x sin x, x 0

Show that the x coordinates of the turning points of the curve with equation y = f(x) satisfy the equation tan x = 4

Figure 3 shows a sketch of part of the curve with equation y = f(x).

Sketch the graph of H against t where

showing the long-term behaviour of this curve.

The function H(t) is used to model the height, in metres, of a ball above the ground t seconds after it has been kicked.

Using this model, find

the maximum height of the ball above the ground between the first and second bounce.

Explain why this model should not be used to predict the time of each bounce.

The value, £ V, of a vintage car t years after it was first valued on 1st January 2001, is modelled by the equation

V=Ap^twhere A and p are constants

Given that the value of the car was £32 000 on 1st January 2005 and £50 000 on 1st January 2012

find p to 4 decimal places,

show that A is approximately 24 800

With reference to the model, interpret

the value of the constant A,

the value of the constant p.

Using the model,

find the year during which the value of the car first exceeds £100 000

Figure 1 is a graph showing the trajectory of a rugby ball.

The height of the ball above the ground, H metres, has been plotted against the horizontal distance, x metres, measured from the point where the ball was kicked.

The ball travels in a vertical plane.

The ball reaches a maximum height of 12 metres and hits the ground at a point 40 metres from where it was kicked.

Find a quadratic equation linking H with x that models this situation.

The ball passes over the horizontal bar of a set of rugby posts that is perpendicular to the path of the ball. The bar is 3 metres above the ground.

Use your equation to find the greatest horizontal distance of the bar from O.

Give one limitation of the model.