Integrating hyperbolic & inverse functions

In this question you must show detailed reasoning.

The power output, p watts, of a machine at time t hours after it is switched on can be modelled by the equation p = 20 - 20 tanh (1.44 t) for t ⩾ 0.

Determine, according to the model, the mean power output of the machine over the first half hour

after it is switched on. Give your answer correct to 2 decimal places.

In this question you must show detailed reasoning.

A pendant is to be made out of a precious metal. The shape of the pendant is modelled as the shape formed when R is rotated by 2 π radians about the x-axis.

Find the exact value of the volume of precious metal required to make the pendant, according to

the model.

A surge in the current, I units, through an electrical component at a time, t seconds, is to be modelled. The surge starts when t = 0 and there is initially no current through the component. When the current has surged for 1 second it is measured as being 5 units. While the surge is occurring, I is modelled by the following differential equation.

By using an integrating factor show that, according to the model, while the surge is occurring,

I is given by I = (2t - t²)(sin^-1(t - 1) + 5).

The surge lasts until there is again no current through the component.

Determine the length of time that the surge lasts according to the model.

Determine, according to the model, the rate of increase of the current at the start of the surge.

Give your answer in an exact form.

In this question you must show detailed reasoning.

form a + ln b + c π where a is an integer, and b and c are both rational.

Using exponentials, show that cosh 2u ≡ 2 sinh²u + 1.

By differentiating both sides of the identity in part (a) with respect to u, show that

sinh 2u ≡ 2 sinh u cosh u.