Separation of variables

Use the substitution u = e + 1 to show that

Solve the differential equation

given that y = π when x = 0

Write your answer in the form

where A and B are constants to be found.

A gardener stores rainwater in a cylindrical container.

The container has a height of 130 centimetres.

The gardener empties the water from the container through a hose.

The hose is attached 5 centimetres from the bottom of the container.

At time t minutes after the hose is switched on, the depth of water, h centimetres, in the container decreases at a rate which is proportional to h – 5

Initially the container of water is full, and the depth of water is decreasing at a rate of 1.5 centimetres per minute.

Show that

Solve the differential equation

to find an expression for h in terms of t

Find the time taken for the container to be half empty.

Give your answer to the nearest minute.

Given that

find the values of A and B

An empty container, in the shape of a cuboid, has length 1.6 metres, width 1.25 metres and depth 0.5 metres, as shown in the diagram below.

The container has a small hole in the bottom.

Water is poured into the container at a rate of 0.16 cubic metres per minute.

At time t minutes after the container starts to be filled, the depth of water is d metres and water leaks out at a rate of 036d² cubic metres per minute.

At time t minutes after the container starts to be filled, the volume of water in the container is V cubic metres.

Show that

Hence, find t in terms of V

Determine how long it takes to fill the container with water.

Give your answer to the nearest minute.

A gardener has a greenhouse containing 900 tomato plants.

The gardener notices that some of the tomato plants are damaged by insects.

Initially there are 25 damaged tomato plants.

The number of tomato plants damaged by insects is increasing by 32% each day.

The total number of plants damaged by insects, x, is modelled by

where A and B are constants and t is the number of days after the gardener first noticed the damaged plants.

Use this model to find the total number of plants damaged by insects 5 days after the gardener noticed the damaged plants.

Explain why this model is not realistic in the long term.

A refined model assumes the rate of increase of the number of plants damaged by insects is given by

Show that

where A and B are positive integers to be found.

Hence, find t in terms of x.

Hence, find the number of days it takes from when the damage is first noticed until half of the plants are damaged by the insects.

Section B

all questions in the spaces provided.

A curve, C, passes through the point with coordinates (1, 6)

The gradient of C is given by

Show that C intersects the coordinate axes at exactly one point and state the coordinates of this point.

Fully justify your answer.

A ball is released from a great height so that it falls vertically downwards towards the surface of the Earth.

Using a simple model, Andy predicts that the velocity of the ball, exactly 2 seconds

Show how Andy has obtained his prediction.

t seconds after being released from rest is

Find an expression for v in terms of t.

Comment on the value of v for the two models as t becomes large.

The initial velocity of the particle is 4 m s¹

By first forming a suitable differential equation, show that

Find the acceleration of the particle when t 55

Solve the differential equation

Write your answer in the form t² f (x)

A market trader notices that daily sales are dependent on two variables:

number of hours, t, after the stall opens

total sales, x, in pounds since the stall opened.

The trader models the rate of sales as directly proportional to

After two hours the rate of sales is £72 per hour and total sales are £336

Show that

Hence, show that

The stall opens at 09.30.

The trader closes the stall when the rate of sales falls below £24 per hour.

Using the results in parts (a) and (b), calculate the earliest time that the trader closes the stall.

Explain why the model used by the trader is not valid at 09.30.

Turn over for Section B

Section B

all questions in the spaces provided.