Forming differential equations

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.

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.

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 new smartphone was released by a company. The company monitored the total number of phones sold, n, at time t days after the phone was released. The company observed that, during this time, the rate of increase of n was proportional to n Use this information to write down a suitable equation for n in terms of t. (You do not need to evaluate any unknown constants in your equation.)

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.

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 vase with a circular cross-section is shown in Figure 2. Water is flowing into the vase. When the depth of the water is h cm, the volume of water V cm³ is given by

Water flows into the vase at a constant rate of 80 cm³s^–1 Find the rate of change of the depth of the water, in cms^–1, when h = 6

At time t seconds the radius of a sphere is r cm, its volume is V cm³ and its surface area is S cm². [You are given that V = r₃ and that S = 4r₂] The volume of the sphere is increasing uniformly at a constant rate of 3 cm³ s^–1.

Find when the radius of the sphere is 4 cm, giving your answer to 3 significant figures.

Find the rate at which the surface area of the sphere is increasing when the radius is 4 cm.

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)

Figure 1 shows a metal cube which is expanding uniformly as it is heated. At time t seconds, the length of each edge of the cube is x cm, and the volume of the cube is V cm³.

Given that the volume, V cm³, increases at a constant rate of 0.048 cm³s^–1,

find the rate of increase of the total surface area of the cube, in cm²s^–1, when x = 8

A hollow hemispherical bowl is shown in Figure 1. Water is flowing into the bowl. When the depth of the water is h m, the volume V m³ is given by

Water flows into the bowl at a rate of m s.

Find the rate of change of h, in m s, when h = 0.1

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

The area A of a circle is increasing at a constant rate of 1.5 cm² s^–1. Find, to 3 significant figures, the rate at which the radius r of the circle is increasing when the area of the circle is 2 cm².

A container is made in the shape of a hollow inverted right circular cone. The height of the container is 24 cm and the radius is 16 cm, as shown in Figure 2. Water is flowing into the container. When the height of water is h cm, the surface of the water has radius r cm and the volume of water is V cm³.

[ The volume V of a right circular cone with vertical height h and base radius r is given

Water flows into the container at a rate of 8 cm³ s^–1.

Find, in terms of π, the rate of change of h when h = 12.

Figure 2 shows a right circular cylindrical metal rod which is expanding as it is heated. After t seconds the radius of the rod is x cm and the length of the rod is 5 x cm. The cross-sectional area of the rod is increasing at the constant rate of 0.032 cm² s^–1.

Find when the radius of the rod is 2 cm, giving your answer to 3 significant figures.

Find the rate of increase of the volume of the rod when x = 2.

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.