To find the points where the height [tex]\( h(t) \)[/tex] is at a maximum, we need to analyze the given equation:
[tex]\[
h(t) = 0.5 \sin \left(\pi t + \frac{\pi}{2}\right) + 1
\][/tex]
First, we look at the sine function [tex]\( \sin \left(\pi t + \frac{\pi}{2}\right) \)[/tex]. The sine function oscillates between -1 and 1. To find the maximum height, we need:
[tex]\[
\sin \left(\pi t + \frac{\pi}{2}\right) = 1
\][/tex]
The sine function [tex]\( \sin(x) \)[/tex] equals 1 at:
[tex]\[
x = \frac{\pi}{2} + 2k\pi \quad \text{for integers } k
\][/tex]
So for the argument of the sine function, we have:
[tex]\[
\pi t + \frac{\pi}{2} = \frac{\pi}{2} + 2k\pi
\][/tex]
Solving for [tex]\( t \)[/tex]:
[tex]\[
\pi t + \frac{\pi}{2} = \frac{\pi}{2} + 2k\pi
\][/tex]
[tex]\[
\pi t = 2k\pi
\][/tex]
[tex]\[
t = 2k \quad \text{for integers } k
\][/tex]
Thus, [tex]\( t = 0, 2, 4, 6, \ldots \)[/tex].
For each of these [tex]\( t \)[/tex]-values, the corresponding height [tex]\( h(t) \)[/tex] is:
[tex]\[
h(t) = 0.5 \times 1 + 1 = 1.5
\][/tex]
Therefore, the points where the height [tex]\( h(t) \)[/tex] is at a maximum are:
[tex]\[
(0, 1.5), (2, 1.5), (4, 1.5), (6, 1.5), \ldots
\][/tex]
On the graph, plot the points [tex]\( (0, 1.5) \)[/tex], [tex]\( (2, 1.5) \)[/tex], [tex]\( (4, 1.5) \)[/tex], [tex]\( (6, 1.5) \)[/tex], and so on.
