Activity 2.4.5.
Consider a spring-mass system where the weight is hanging from the ceiling in such a way that the following is known: we let \(d(t)\) denote the distance from the ceiling to the weight at time \(t\) in seconds and know that the weight oscillates periodically with a minimum value of \(1.5\) feet and a maximum value of \(4\) feet, with a period of \(3\text{,}\) and you know \(d(0.5) = 2.75\) and \(d\left(1.25\right) = 4\text{.}\)
State the midline, amplitude, range, and an anchor point for the function, and hence determine a formula for \(d(t)\) in the form \(a\cos(k(t-b))+c\) or \(a\sin(k(t-b))+c\text{.}\) Show your work and thinking, and use Desmos appropriately to check that your formula generates the desired behavior.
