Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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].
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
