Separation of variables

Show that the equation of the curve can be written in the form cos y = f (x), where f (x) is

fully simplified.

Find the equation of this curve, giving your answer in the form e^3y = f(x).

Show that, when x = e², the y-coordinate of this curve can be written as

y = a + ln(be³ + c), where a, b and c are constants to be determined.

The diagram shows a water tank which is shaped as an inverted cone with semi-vertical angle 30° and height 50 cm. Initially the tank is full, and the depth of the water is 50 cm.

Water flows out of a small hole at the bottom of the tank. The rate at which the water flows out is

Determine the time taken for the tank to become empty.

[For a cone with base radius r and height h the volume V is given by r²h.]

A cake is cooling so that, t minutes after it is removed from an oven, its temperature is θ °C. When the cake is removed from the oven, its temperature is 160 °C. After 10 minutes its temperature has fallen to 125 °C.

In a simple model, the rate of decrease of the temperature of the cake is assumed to be constant.

Write down a differential equation for this model.

Solve this differential equation to find θ in terms of t.

State one limitation of this model.

In a revised model, the rate of decrease of the temperature of the cake is proportional to the difference between the temperature of the cake and the temperature of the room. The temperature of the room is a constant 20 °C.

Write down a differential equation for this revised model.

Solve this differential equation to find θ in terms of t.

The cake can be decorated when its temperature is 25 °C. Find the difference in time between when the two models would predict that the cake can be decorated, giving your answer correct

to the nearest minute.

Express this property in the form of a differential equation.

It is given that C passes through the point (4, 3) and that x > 3 and y > 2 at all points on C.

Determine the equation of C giving your answer in the form y = f (x).

The curve C may be obtained by a transformation of part of the curve y = x³.

Describe fully this transformation.

Determine an expression for P in terms of t.

Describe how the population changes over time.

A cylindrical tank is initially full of water. There is a small hole at the base of the tank out of which the water leaks.

The height of water in the tank is x m at time t seconds. The rate of change of the height of water may be modelled by the assumption that it is proportional to the square root of the height of water.

When t = 100, x = 0.64 and, at this instant, the height is decreasing at a rate of 0.0032ms^-1.

Find an expression for x in terms of t.

Hence determine at what time, according to this model, the tank will be empty.