ABSTRACT
Problem 3. Newtons law of cooling is given by: D 0ekt, where the excess of temperature at zero time is 0 °C and at time t seconds is °C. Determine the rate of change of temperature after 40 s, given that 0 D 16 °C and k D 0.03
The rate of change of temperature is d d t
Since D 0ekt then dd t D 0ke kt
D k0ekt When 0 D 16, k D 0.03 and t D 40 then
d d t
D 0.0316e0.0340
D 0.48 e1.2 D 1.594 °C=s
Problem 4. The displacement s cm of the end of a stiff spring at time t seconds is given by: s D aekt sin 2ft. Determine the velocity of the end of the spring after 1 s, if a D 2, k D 0.9 and f D 5
Velocity v D ds d t
where s D aekt sin 2ft (i.e. a product)
Using the product rule, ds d t
D aekt2f cos 2ft
When a D 2, k D 0.9, f D 5 and t D 1,
velocity, v D 2e0.925 cos 25 C sin 2520.9e0.9
D 25.5455 cos 10 0.7318 sin 10 D 25.54551 0.73180 D 25.55 cm=s
(Note that cos 10 means ‘the cosine of 10 radians’, not degrees, and cos 10 cos 2 D 1).
