To find the points where the height [tex]\( h(t) \)[/tex] of the weight on the spring is at a maximum, follow these steps:
1. Understand the given function:
The height function given is:
[tex]\[
h(t) = 0.5 \cdot \sin \left(\pi t + \frac{\pi}{2}\right) + 1
\][/tex]
2. Analyze the sine function:
The sine function [tex]\(\sin(\theta)\)[/tex] has a maximum value of 1. We need to find the values of [tex]\( t \)[/tex] for which:
[tex]\[
\sin\left(\pi t + \frac{\pi}{2}\right) = 1
\][/tex]
This equation is true when:
[tex]\[
\pi t + \frac{\pi}{2} = \frac{\pi}{2} + 2k\pi \quad \text{where} \; k \; \text{is an integer}
\][/tex]
Simplifying gives:
[tex]\[
\pi t = 2k\pi \implies t = 2k
\][/tex]
3. Identify specific maximum points:
Some specific values of [tex]\( t \)[/tex] where [tex]\( t = 2k \)[/tex] would be:
[tex]\[
t = 0, 2, 4, \ldots
\][/tex]
4. Calculate the maximum height:
Substituting these [tex]\( t \)[/tex] values back into the height function [tex]\( h(t) \)[/tex], we get:
[tex]\[
h(t) = 0.5 \cdot 1 + 1 = 1.5
\][/tex]
5. Maximum height points:
Therefore, the points where height [tex]\( h(t) \)[/tex] is at a maximum are:
[tex]\[
(0, 1.5), (2, 1.5), (4, 1.5), \ldots
\][/tex]
Using these steps, you should plot the points (0, 1.5), (2, 1.5), (4, 1.5), etc., on the given graph to show the points where the weight is at its maximum height above the ground.
