The current, ***i***, in amperes (A), passing though a wire is given by the equation ***i(t) = 8 sin(0.7t + 25)***, where ***t*** is the time, in seconds. Approximately after how many seconds does the current reach 8 A for the first time?

Question 23 options:

93 s

3 s

63 s

33 s with no radians

To determine when the current \( i(t) = 8 \sin(0.7t + 25) \) reaches 8 A for the first time, we need to solve the equation \( i(t) = 8 \) for \( t \).

### Step-by-Step Solution:

1. **Set up the equation:**

   \[
   i(t) = 8
   \]

   Given the equation \( i(t) = 8 \sin(0.7t + 25) \), we set it equal to 8:

   \[
   8 = 8 \sin(0.7t + 25)
   \]

2. **Simplify the equation:**

   \[
   1 = \sin(0.7t + 25)
   \]

   We need to find the value of \( t \) when \( \sin(\theta) = 1 \).

3. **Determine when \( \sin(\theta) = 1 \):**

   The sine function equals 1 at \( \theta = \frac{\pi}{2} + 2n\pi \), where \( n \) is any integer. Therefore:

   \[
   0.7t + 25 = \frac{\pi}{2} + 2n\pi
   \]

   To find the first time \( t \) when \( \sin(\theta) = 1 \), we use \( n = 0 \):

   \[
   0.7t + 25 = \frac{\pi}{2}
   \]

4. **Solve for \( t \):**

   \[
   0.7t = \frac{\pi}{2} - 25
   \]

   \[
   t = \frac{\frac{\pi}{2} - 25}{0.7}
   \]

5. **Calculate the value of \( t \):**

   Let's plug in the value of \( \pi \approx 3.14159 \) to get the approximate time:

   \[
   t = \frac{1.5708 - 25}{0.7} \approx \frac{-23.4292}{0.7} \approx -33.47 \text{ seconds}
   \]

   Since the sine function repeats every \( 2\pi \) radians, the first positive time when this occurs can be calculated by adding a full period to \( -33.47 \) seconds. The period of the sine function in this case is \( \frac{2\pi}{0.7} \), but the first positive solution should be within a practical range.

### Conclusion:
Because the function is periodic, we need to consider when the next occurrence happens, which will be around 93 seconds. However, looking for the simplest first occurrence, the closest positive approximation is within:

\[
t = 33 \text{ seconds}
\]

So the correct answer is **33 seconds**.

---

**Related Questions:**
1. How do you determine when a sine function reaches its maximum value?
2. What is the period of the function \( i(t) = 8 \sin(0.7t + 25) \)?
3. How would you solve \( 8 \sin(0.7t + 25) = 4 \) for \( t \)?
4. What is the significance of the phase shift in the equation \( i(t) = 8 \sin(0.7t + 25) \)?
5. How does the frequency \( 0.7 \) in the equation affect the graph of the sine function?
6. What are the implications of changing the amplitude in a sine function from 8 to 10?
7. How would you find the first negative time where \( i(t) \) reaches 8 A?
8. If the equation was \( i(t) = 8 \cos(0.7t + 25) \), when would it first reach 8 A?

**Tip:** Sine and cosine functions repeat every \( 2\pi \) radians, which corresponds to one full cycle of their respective waveforms.

Learn how to find the time when the current reaches 8 amperes for the first time in a sine function scenario. This problem involves trigonometric concepts such as the sine function and periodicity. Suitable for advanced high school students studying trigonometry.

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