Answer :
To solve the problem with the given function \( h(t) = 53 + 50 \sin \left(\frac{\pi}{10} t - \frac{\pi}{2}\right) \), we need to evaluate this function at specific values of \( t \). Let's take the values \( t = 0 \), \( t = 5 \), \( t = 10 \), \( t = 15 \), and \( t = 20 \).
1. For \( t = 0 \):
[tex]\[ h(0) = 53 + 50 \sin \left(\frac{\pi}{10} \cdot 0 - \frac{\pi}{2}\right) \][/tex]
Simplifying inside the sine function:
[tex]\[ h(0) = 53 + 50 \sin \left(-\frac{\pi}{2}\right) \][/tex]
The value of \(\sin\left(-\frac{\pi}{2}\right)\) is \(-1\):
[tex]\[ h(0) = 53 + 50 \cdot (-1) = 53 - 50 = 3 \][/tex]
2. For \( t = 5 \):
[tex]\[ h(5) = 53 + 50 \sin \left(\frac{\pi}{10} \cdot 5 - \frac{\pi}{2}\right) \][/tex]
Simplifying inside the sine function:
[tex]\[ h(5) = 53 + 50 \sin \left(\frac{5\pi}{10} - \frac{\pi}{2}\right) = 53 + 50 \sin \left(\frac{\pi}{2} - \frac{\pi}{2}\right) \][/tex]
The value of \(\sin(0)\) is 0:
[tex]\[ h(5) = 53 + 50 \cdot 0 = 53 \][/tex]
3. For \( t = 10 \):
[tex]\[ h(10) = 53 + 50 \sin \left(\frac{\pi}{10} \cdot 10 - \frac{\pi}{2}\right) \][/tex]
Simplifying inside the sine function:
[tex]\[ h(10) = 53 + 50 \sin \left(\pi - \frac{\pi}{2}\right) = 53 + 50 \sin \left(\frac{\pi}{2}\right) \][/tex]
The value of \(\sin\left(\frac{\pi}{2}\right)\) is 1:
[tex]\[ h(10) = 53 + 50 \cdot 1 = 53 + 50 = 103 \][/tex]
4. For \( t = 15 \):
[tex]\[ h(15) = 53 + 50 \sin \left(\frac{\pi}{10} \cdot 15 - \frac{\pi}{2}\right) \][/tex]
Simplifying inside the sine function:
[tex]\[ h(15) = 53 + 50 \sin \left(\frac{15\pi}{10} - \frac{\pi}{2}\right) = 53 + 50 \sin \left(\frac{3\pi}{2} - \frac{\pi}{2}\right) \][/tex]
Which simplifies to:
[tex]\[ h(15) = 53 + 50 \sin \left(\pi\right) \][/tex]
The value of \(\sin(\pi)\) is 0:
[tex]\[ h(15) = 53 + 50 \cdot 0 = 53 \][/tex]
5. For \( t = 20 \):
[tex]\[ h(20) = 53 + 50 \sin \left(\frac{\pi}{10} \cdot 20 - \frac{\pi}{2}\right) \][/tex]
Simplifying inside the sine function:
[tex]\[ h(20) = 53 + 50 \sin \left(2\pi - \frac{\pi}{2}\right) = 53 + 50 \sin \left(\frac{3\pi}{2}\right) \][/tex]
The value of \(\sin\left( \frac{3\pi}{2} \right)\) is \(-1\):
[tex]\[ h(20) = 53 + 50 \cdot (-1) = 53 - 50 = 3 \][/tex]
Hence, the results for \( h(t) \) at the given values of \( t \) are:
- \( h(0) = 3 \)
- \( h(5) = 53 \)
- \( h(10) = 103 \)
- \( h(15) = 53 \)
- \( h(20) = 3 \)
These results are [tex]\([3, 53, 103, 53, 3]\)[/tex].
1. For \( t = 0 \):
[tex]\[ h(0) = 53 + 50 \sin \left(\frac{\pi}{10} \cdot 0 - \frac{\pi}{2}\right) \][/tex]
Simplifying inside the sine function:
[tex]\[ h(0) = 53 + 50 \sin \left(-\frac{\pi}{2}\right) \][/tex]
The value of \(\sin\left(-\frac{\pi}{2}\right)\) is \(-1\):
[tex]\[ h(0) = 53 + 50 \cdot (-1) = 53 - 50 = 3 \][/tex]
2. For \( t = 5 \):
[tex]\[ h(5) = 53 + 50 \sin \left(\frac{\pi}{10} \cdot 5 - \frac{\pi}{2}\right) \][/tex]
Simplifying inside the sine function:
[tex]\[ h(5) = 53 + 50 \sin \left(\frac{5\pi}{10} - \frac{\pi}{2}\right) = 53 + 50 \sin \left(\frac{\pi}{2} - \frac{\pi}{2}\right) \][/tex]
The value of \(\sin(0)\) is 0:
[tex]\[ h(5) = 53 + 50 \cdot 0 = 53 \][/tex]
3. For \( t = 10 \):
[tex]\[ h(10) = 53 + 50 \sin \left(\frac{\pi}{10} \cdot 10 - \frac{\pi}{2}\right) \][/tex]
Simplifying inside the sine function:
[tex]\[ h(10) = 53 + 50 \sin \left(\pi - \frac{\pi}{2}\right) = 53 + 50 \sin \left(\frac{\pi}{2}\right) \][/tex]
The value of \(\sin\left(\frac{\pi}{2}\right)\) is 1:
[tex]\[ h(10) = 53 + 50 \cdot 1 = 53 + 50 = 103 \][/tex]
4. For \( t = 15 \):
[tex]\[ h(15) = 53 + 50 \sin \left(\frac{\pi}{10} \cdot 15 - \frac{\pi}{2}\right) \][/tex]
Simplifying inside the sine function:
[tex]\[ h(15) = 53 + 50 \sin \left(\frac{15\pi}{10} - \frac{\pi}{2}\right) = 53 + 50 \sin \left(\frac{3\pi}{2} - \frac{\pi}{2}\right) \][/tex]
Which simplifies to:
[tex]\[ h(15) = 53 + 50 \sin \left(\pi\right) \][/tex]
The value of \(\sin(\pi)\) is 0:
[tex]\[ h(15) = 53 + 50 \cdot 0 = 53 \][/tex]
5. For \( t = 20 \):
[tex]\[ h(20) = 53 + 50 \sin \left(\frac{\pi}{10} \cdot 20 - \frac{\pi}{2}\right) \][/tex]
Simplifying inside the sine function:
[tex]\[ h(20) = 53 + 50 \sin \left(2\pi - \frac{\pi}{2}\right) = 53 + 50 \sin \left(\frac{3\pi}{2}\right) \][/tex]
The value of \(\sin\left( \frac{3\pi}{2} \right)\) is \(-1\):
[tex]\[ h(20) = 53 + 50 \cdot (-1) = 53 - 50 = 3 \][/tex]
Hence, the results for \( h(t) \) at the given values of \( t \) are:
- \( h(0) = 3 \)
- \( h(5) = 53 \)
- \( h(10) = 103 \)
- \( h(15) = 53 \)
- \( h(20) = 3 \)
These results are [tex]\([3, 53, 103, 53, 3]\)[/tex].
