Partial fractions

The expression

can be written in the form

Find the value of A and the value of B

Use your answer to part (a) to show that

where q is a rational number to be found.

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.

Express

in the form

where A and B are integers.

Show that

where p and q are rational numbers.