Integrating factor

A rambler starts a walk at the bottom of a hill at 10 am.

The rambler walks all the way to the top of the hill and then turns around and walks back down to the bottom of the hill.

The differential equation

where A is a constant, is used to model the vertical displacement, x metres, of the rambler from the bottom of the hill, t hours after the start of the walk.

Given that after 1 hour

• the rambler has a vertical displacement of 213 m

determine the value of A to 3 significant figures.

Hence, determine the particular solution of the differential equation, giving your answer in the form x = f (t)

Use the model to determine the time at which the rambler will return to the bottom of the hill.

Given that the vertical displacement of the top of the hill is 300.68 m

use the model to find the value of t when the rambler reaches the top of the hill.

Using your answers to parts (c) and (d), give a limitation of the model.

The concentration, Pmg m^–3, of a pollutant in a reservoir, t days after the pollutant entered the reservoir, is modelled by the differential equation

Show that the transformation x = transforms equation (I) into the equation

Given that P = 0.5 when t = 0

solve differential equation (II) to show that, according to the model,

where k is a constant to be determined.

Given that the concentration of the pollutant in the reservoir, 3 days after the pollutant entered the reservoir, was 0.034 mg m^–3

comment on the reliability of the model, giving a reason for your answer.

A raindrop falls from rest from a cloud. The velocity, vm s^–1 vertically downwards, of the raindrop, t seconds after the raindrop starts to fall, is modelled by the differential equation

where k is a positive constant.

Solve the differential equation to show that

Given that v = 4 whent = 2

determine, according to the model, the velocity of the raindrop 5 seconds after it starts to fall.

Comment on the validity of the model for very large values of t

A raindrop falls from rest from a cloud. The velocity, vm s^–1 vertically downwards, of the raindrop, t seconds after the raindrop starts to fall, is modelled by the differential equation

where k is a positive constant.

Solve the differential equation to show that

Given that v = 4 whent = 2

determine, according to the model, the velocity of the raindrop 5 seconds after it starts to fall.

Comment on the validity of the model for very large values of t

Determine the general solution of the differential equation

giving your answer in the form y = f(x)

Given that y = 3 when x = 0

determine the smallest positive value of x for which y = 0

Determine the general solution of the differential equation

giving your answer in the form y = f(x)

Given that y = 3 when x = 0

determine the smallest positive value of x for which y = 0

A community is concerned about the rising level of pollutant in its local pond and applies a chemical treatment to stop the increase of pollutant.

The concentration, x parts per million (ppm), of the pollutant in the pond water t days after the chemical treatment was applied, is modelled by the differential equation

When the chemical treatment was applied the concentration of pollutant was 3 ppm.

Use the iteration formula

once to estimate the concentration of the pollutant in the pond water 6 hours after the chemical treatment was applied.

Show that the transformation u = x³ transforms the differential equation (I) into the differential equation

Determine the general solution of equation (II)

Hence find an equation for the concentration of pollutant in the pond water t days after the chemical treatment was applied.

Find the percentage error of the estimate found in part (a) compared to the value predicted by the model, stating if it is an overestimate or an underestimate.

A sample of bacteria in a sealed container is being studied.

The number of bacteria, P, in thousands, is modelled by the differential equation

where t is the time in hours after the start of the study.

Initially, there are exactly 5000 bacteria in the container.

Determine, according to the model, the number of bacteria in the container 8 hours after the start of the study.

Find, according to the model, the rate of change of the number of bacteria in the container 4 hours after the start of the study.

State a limitation of the model.

A sample of bacteria in a sealed container is being studied.

The number of bacteria, P, in thousands, is modelled by the differential equation

where t is the time in hours after the start of the study.

Initially, there are exactly 5000 bacteria in the container.

Determine, according to the model, the number of bacteria in the container 8 hours after the start of the study.

Find, according to the model, the rate of change of the number of bacteria in the container 4 hours after the start of the study.

State a limitation of the model.

A tank at a chemical plant has a capacity of 250 litres. The tank initially contains 100 litres of pure water.

Salt water enters the tank at a rate of 3 litres every minute. Each litre of salt water entering the tank contains 1 gram of salt.

It is assumed that the salt water mixes instantly with the contents of the tank upon entry.

At the instant when the salt water begins to enter the tank, a valve is opened at the bottom of the tank and the solution in the tank flows out at a rate of 2 litres per minute.

Given that there are S grams of salt in the tank after t minutes,

=show that the situation can be modelled by the differential equation

Hence find the number of grams of salt in the tank after 10 minutes.

When the concentration of salt in the tank reaches 0.9 grams per litre, the valve at the bottom of the tank must be closed.

=Find, to the nearest minute, when the valve would need to be closed.

Evaluate the model.

A tank at a chemical plant has a capacity of 250 litres. The tank initially contains 100 litres of pure water.

Salt water enters the tank at a rate of 3 litres every minute. Each litre of salt water entering the tank contains 1 gram of salt.

It is assumed that the salt water mixes instantly with the contents of the tank upon entry.

At the instant when the salt water begins to enter the tank, a valve is opened at the bottom of the tank and the solution in the tank flows out at a rate of 2 litres per minute.

Given that there are S grams of salt in the tank after t minutes,

show that the situation can be modelled by the differential equation

Hence find the number of grams of salt in the tank after 10 minutes.

When the concentration of salt in the tank reaches 0.9 grams per litre, the valve at the bottom of the tank must be closed.

Find, to the nearest minute, when the valve would need to be closed.

Evaluate the model.