Separation of variables

Water flows at a constant rate into a large container. There is a tap at the bottom of the container. At time t hours after the tap was opened

• the volume of water in the container is Vm³

• water is flowing into the container at a constant rate of 0.45 m³ per hour

• water is leaving the container through the tap at a rate of 0.3Vm³ per hour

Show that

Given that when the tap was opened, there was 0.25 m³ of water in the container,

solve the differential equation to show that

where P, Q and k are positive constants to be found. Given that

• the capacity of the container is 2 m³

• the tap remains open

• the water continues to flow into the tank at the same rate

determine whether the container will ever become full, giving a reason for your answer.

Figure 2 shows a cylindrical tank of height 1.5 m.

Initially the tank is full of water.

The water starts to leak from a small hole, at a point L, in the side of the tank.

While the tank is leaking, the depth, H metres, of the water in the tank is modelled by the differential equation

where t hours is the time after the leak starts.

Using the model,

show that

where A and B are constants to be found,

find the time taken for the depth of the water to decrease to 1.2 m. Give your answer in hours and minutes, to the nearest minute.

In the long term, the water level in the tank falls to the same height as the hole.

Find, according to the model, the height of the hole from the bottom of the tank.

A balloon is being inflated.

In a simple model,

• the balloon is modelled as a sphere

• the rate of increase of the radius of the balloon is inversely proportional to the square root of the radius of the balloon

At time t seconds, the radius of the balloon is rcm.

Write down a differential equation to model this situation.

At the instant when t = 10

• the radius is 16 cm

• the radius is increasing at a rate of 0.9 cm s^–1

Solve the differential equation to show that

Hence find the radius of the balloon when t = 20

Give your answer to the nearest millimetre.

Suggest a limitation of the model.

The volume, V microlitres, of a plant cell t hours after the plant is watered is modelled by the differential equation

The plant cell has an initial volume of 20 microlitres.

Find, according to the model, the time taken, in minutes, for the volume of the plant cell to reach 24 microlitres.

Show that

where A, B and k are constants to be found.

The model predicts that there is an upper limit, L microlitres, on the volume of the plant cell.

Find the value of L, giving a reason for your answer.

A tank in the shape of a cuboid is being filled with water. The base of the tank measures 20 m by 10 m and the height of the tank is 5 m, as shown in Figure 1. At time t minutes after water started flowing into the tank the height of the water was hm and the volume of water in the tank was Vm³ In a model of this situation

• the sides of the tank have negligible thickness

• the rate of change of V is inversely proportional to the square root of h

Show that

where λ is a constant. Given that

• initially the height of the water in the tank was 1.44 m

• exactly 8 minutes after water started flowing into the tank the height of the water was 3.24 m

use the model to find an equation linking h with t, giving your answer in the form

where A and B are constants to be found.

Hence find the time taken, from when water started flowing into the tank, for the tank to be completely full.

Water flows at a constant rate into a large tank. The tank is a cuboid, with all sides of negligible thickness. The base of the tank measures 8 m by 3 m and the height of the tank is 5 m. There is a tap at a point T at the bottom of the tank, as shown in Figure 5. At time t minutes after the tap has been opened

• the depth of water in the tank is h metres

• water is flowing into the tank at a constant rate of 0.48 m³ per minute

• water is modelled as leaving the tank through the tap at a rate of 0.1hm³ per minute

Show that, according to the model,

Given that when the tap was opened, the depth of water in the tank was 2 m,

show that, according to the model,

where A, B and k are constants to be found. Given that the tap remains open,

determine, according to the model, whether the tank will ever become full, giving a reason for your answer.

Water flows at a constant rate into a large tank.

The tank is a cuboid, with all sides of negligible thickness.

The base of the tank measures 8 m by 3 m and the height of the tank is 5 m.

There is a tap at a point T at the bottom of the tank, as shown in Figure 5.

At time t minutes after the tap has been opened

• the depth of water in the tank is h metres

• water is flowing into the tank at a constant rate of 0.48 m³ per minute

• water is modelled as leaving the tank through the tap at a rate of 0.1hm³ per minute

Show that, according to the model,

Given that when the tap was opened, the depth of water in the tank was 2 m,

show that, according to the model,

where A, B and k are constants to be found.

Given that the tap remains open,

determine, according to the model, whether the tank will ever become full, giving a reason for your answer.

A large spherical balloon is deflating. At time t seconds the balloon has radius rcm and volume Vcm³ The volume of the balloon is modelled as decreasing at a constant rate.

Using this model, show that

where k is a positive constant. Given that ● the initial radius of the balloon is 40 cm ● after 5 seconds the radius of the balloon is 20 cm ● the volume of the balloon continues to decrease at a constant rate until the balloon is empty

solve the differential equation to find a complete equation linking r and t.

Find the limitation on the values of t for which the equation in part (b) is valid.

where k is a constant

A team of scientists is studying a species of slow growing tree. The rate of change in height of a tree in this species is modelled by the differential equation

where h is the height in metres and t is the time, measured in years, after the tree is planted.

Find, according to the model, the range in heights of trees in this species.

One of these trees is one metre high when it is first planted. According to the model,

calculate the time this tree would take to reach a height of 12 metres, giving your answer to 3 significant figures.

The height above ground, H metres, of a passenger on a roller coaster can be modelled by the differential equation

where t is the time, in seconds, from the start of the ride.

Given that the passenger is 5 m above the ground at the start of the ride,

State the maximum height of the passenger above the ground.

The passenger reaches the maximum height, for the second time, T seconds after the start of the ride.

Find the value of T.

A spherical mint of radius 5 mm is placed in the mouth and sucked.

Four minutes later, the radius of the mint is 3 mm.

In a simple model, the rate of decrease of the radius of the mint is inversely proportional to the square of the radius.

Using this model and all the information given,

find an equation linking the radius of the mint and the time. (You should define the variables that you use.)

Hence find the total time taken for the mint to completely dissolve. Give your answer in minutes and seconds to the nearest second.

Suggest a limitation of the model.

A container with a circular cross-section is shown in Figure 3. Initially the container is empty. At time t seconds after water begins to flow into the container, the height of water in the container is h cm. The height of water in the container satisfies the differential equation

where k is a constant. When h = 16, the height of water in the container is increasing at a rate of 0.6 cm s

Find the value of k.

Find the time taken to fill the container with water from empty to a height of 30 cm.

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find the volume of water in the container when h = 30 Give your answer in cm³ to 3 significant figures.

Figure 3 shows a vertical cylindrical tank of height 200 cm containing water. Water is leaking from a hole P on the side of the tank. At time t minutes after the leaking starts, the height of water in the tank is h cm. The height h cm of the water in the tank satisfies the differential equation

where k is a constant. Given that, when h = 130, the height of the water is falling at a rate of 1.1cm per minute,

find the value of k.

Given that the tank was full of water when the leaking started,

solve the differential equation with your value of k, to find the value of t when h = 50

The rate of decay of the mass of a particular substance is modelled by the differential equation

where x is the mass of the substance measured in grams and t is the time measured in days. Given that x = 60 when t = 0,

solve the differential equation, giving x in terms of t. You should show all steps in your working and give your answer in its simplest form.

Find the time taken for the mass of the substance to decay from 60 grams to 20 grams. Give your answer to the nearest minute.

A team of biologists is studying a population of a particular species of animal. The population is modelled by the differential equation

where P is the population in thousands, and t is the time measured in years since the start of the study. Given that P = 3 when t = 0,

solve this differential equation to show that

find the time taken for the population to reach 4000 for the first time. Give your answer in years to 3 significant figures.

Find

Find

Given that y = at x = 0, solve the differential equation

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The rate of increase of the number, N, of fish in a lake is modelled by the differential equation

In the given equation, the time t is measured in years from the start of January 2000 and k is a positive constant.

By solving the differential equation, show that

where A is a positive constant.

After one year, at the start of January 2001, there are 1200 fish in the lake. After two years, at the start of January 2002, there are 1800 fish in the lake.

Find the exact value of the constant A and the exact value of the constant k.

Hence find the number of fish in the lake after five years. Give your answer to the nearest hundred fish.

Water is being heated in a kettle. At time t seconds, the temperature of the water is °C. The rate of increase of the temperature of the water at any time t is modelled by the differential equation

where is a positive constant. Given that θ = 20 when t = 0,

solve this differential equation to show that

When the temperature of the water reaches 100 °C, the kettle switches off.

Given that λ = 0.01, find the time, to the nearest second, when the kettle switches off.

In an experiment testing solid rocket fuel, some fuel is burned and the waste products are collected. Throughout the experiment the sum of the masses of the unburned fuel and waste products remains constant. Letx be the mass ofwaste products, in kg, at time ¢ minutes after the start ofthe experiment. It is known that at time ¢ minutes, the rate of increase of the mass of waste products, inconstant.kg per minute, is k times the mass of unburned fuel remaining, where k is a positive

The differential equation connecting x and ¢ may be written in the form

Explain, in the context of the problem, what and M represent. (2)

Given that initially the mass of waste products is zero,

solve the differential equation, expressing x in terms of k, M and ¢. (6)

A bottle of water is put into a refrigerator. The temperature inside the refrigerator remains constant at 3 °C and t minutes after the bottle is placed in the refrigerator the temperature of the water in the bottle is °C. The rate of change of the temperature of the water in the bottle is modelled by the differential equation,

By solving the differential equation, show that,

where A is a constant.

Given that the temperature of the water in the bottle when it was put in the refrigerator was 16 °C,

find the time taken for the temperature of the water in the bottle to fall to 10°C, giving your answer to the nearest minute.

Given that y = 2 at x = solve the differential equation

Given that y 1.5 at x = – 2, solve the differential equation

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

Find the particular solution of the differential equation

Figure 2 shows a cylindrical water tank. The diameter of a circular cross-section of the tank is 6 m. Water is flowing into the tank at a constant rate of 0.48π m³ min¹. At time t minutes, the depth of the water in the tank is h metres. There is a tap at a point T at the bottom of the tank. When the tap is open, water leaves the tank at a rate of 0.6πh m³ min¹.

Show that t minutes after the tap has been opened

When t 0, h 0.2

Find the value of t when h 0.5

Given that y = 8 at x =1, solve the differential equation

giving your answer in the form y² = g(x).

Hence obtain the solution of

for which y = 0 at x =x giving your answer in the form sec² x = g(y).

Liquid is pouring into a large vertical circular cylinder at a constant rate of 1600 cm³ s^–1 and is leaking out of a hole in the base, at a rate proportional to the square root of the height of the liquid already in the cylinder. The area of the circular cross section of the cylinder is 4000 cm².

Show that at time t seconds, the height h cm of liquid in the cylinder satisfies the differential equation

When h = 25, water is leaking out of the hole at 400 cm³ s^–1.

Show that k = 0.02

Separate the variables of the differential equation

to show that the time taken to fill the cylinder from empty to a height of 100 cm is given by

Hence find the time taken to fill the cylinder from empty to a height of 100 cm, giving your answer in minutes and seconds to the nearest second.