# Richard Iii

Question 1:

t minutes after a jet engine starts operation, the rate of fuel burn, R kg per minute, is given by the
relation   .
a. Draw a sketch of R as a function of t.
b. What is the rate of burn, R, after 7 minutes?
c. What value does R approach as t becomes very large?
d. Calculate the total amount of fuel burned in the first 7 minutes.¤

Question 2:

The rate at which a body cools in air is assumed to be proportional to the difference between its
temperature T and the constant temperature S of the surrounding air.   This can be expressed by
the differential equation   where t is the time in hours and k is a constant.
a. Show that T = S + Bekt, where B is a constant, is a solution of the differential equation.
b. A heated body cools from 80 C to 40 C in 2 hours.   The air temperature S around the body is 20 C.   Find the temperature of the body after one further hour has elapsed.   Give your answer correct to the nearest degree.¤

Question 3:

The radius of a circular metal plate is increasing at the rate of 0.01 cm/s when subjected to heat.
At what rate is the area increasing when the radius is 5cm?

Question 4:

i. Show that   .
ii. The acceleration of a particle moving in a straight line is given by   where x
metres is the displacement from the origin.   Initially, the particle is at the origin with
velocity 2 ms–1.   Prove that   .

iii. What happens to v as x increases without bound?¤

Question 5:

The path of a projectile fired from the origin O is given by x = Vt cos ,
y = Vt sin  - 5t2 where V is the initial speed in metres per second and  is the angle of projection as in the diagram and t is the time in seconds.
i. Find the maximum height reached by the projectile in terms of V and .
ii. Find the range in terms of V and .
iii. Prove that the range is maximum when  = 45.¤