1.1. The length of a spring, $l$ cm, $t$ seconds after being released from rest is modelled by the equation

$l=a+b\text{cos}4tt\geq0$

where $a$ and $b$ are constants.

According to the model, state what the constants $a$ and $b$ represent in terms of the length of the spring.

1.2. Given that the minimum length of the spring is exactly $12$ cm and the maximum length of the spring is exactly $30$ cm, find the value of $a$ and the value of $b$.

1.3. The length of a similar spring, with the same values of $a$ and $b$, is modelled by the equation

$l=a+b\text{cos}2tt\geq0$

(i) Compare the motion of this spring with the motion of the original spring.

(ii) State one refinement to the models that would make them more realistic.