Question

a student observes that the motion of a weight oscillating up and down on a spring can be modeled by this equation, where (h(t)) is the weight above the ground, in meters, and (t) is the time, in seconds.(h(t)=0.5sin(pi t+\frac{pi}{2}) + 1)on the graph, plot the points where height, (h(t)), is at a maximum.

Explanation:

Step1: Recall sine - function property

The maximum value of the sine function $y = \sin(u)$ is 1. We want to find when $\sin(\pi t+\frac{\pi}{2}) = 1$.

Step2: Solve for $t$

Set $\pi t+\frac{\pi}{2}=\frac{\pi}{2}+ 2k\pi$, where $k\in\mathbb{Z}$ (since the sine - function reaches its maximum at $\frac{\pi}{2}+2k\pi$ for integer values of $k$).
Subtract $\frac{\pi}{2}$ from both sides: $\pi t=2k\pi$.
Divide both sides by $\pi$: $t = 2k$, $k\in\mathbb{Z}$.

Step3: Find the maximum value of $h(t)$

When $\sin(\pi t+\frac{\pi}{2}) = 1$, $h(t)=0.5\times1 + 1=1.5$.
The points where $h(t)$ is at a maximum are of the form $(t, h(t))=(2k,1.5)$ for $k = 0,1,2,\cdots$. For non - negative values of $t$ (since $t$ represents time), when $k = 0$, the point is $(0,1.5)$; when $k = 1$, the point is $(2,1.5)$; when $k = 2$, the point is $(4,1.5)$; when $k = 3$, the point is $(6,1.5)$.

Answer:

The points $(0,1.5),(2,1.5),(4,1.5),(6,1.5)$ (and other points of the form $(2k,1.5)$ for non - negative integers $k$) are the points where the height $h(t)$ is at a maximum.