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To solve for the time \( t \) when the brightness \( B(t) \) is equal to 14, we use the given equation:

\[ B(t) = 7.5 + 10.5 \sin \left( \frac{2\pi}{65} t \right) \]

Given:
\[ B(t) = 14 \]

Substitute \( B(t) = 14 \) into the equation:

\[ 14 = 7.5 + 10.5 \sin \left( \frac{2\pi}{65} t \right) \]

Subtract 7.5 from both sides:

\[ 6.5 = 10.5 \sin \left( \frac{2\pi}{65} t \right) \]

Divide both sides by 10.5:

\[ \sin \left( \frac{2\pi}{65} t \right) = \frac{6.5}{10.5} \]

\[ \sin \left( \frac{2\pi}{65} t \right) = \frac{65}{105} \]

\[ \sin \left( \frac{2\pi}{65} t \right) = \frac{13}{21} \]

Now solve for \( t \):

\[ \frac{2\pi}{65} t = \arcsin \left( \frac{13}{21} \right) \]

First, find the value of \(\arcsin \left( \frac{13}{21} \right)\):

\[ \arcsin \left( \frac{13}{21} \right) \approx 0.6215 \text{ radians} \]

Then:

\[ \frac{2\pi}{65} t = 0.6215 \]

Solve for \( t \):

\[ t = \frac{0.6215 \times 65}{2\pi} \]

\[ t \approx \frac{40.3975}{6.2832} \]

\[ t \approx 6.43 \text{ days} \]

Since the sine function is periodic, we need to check for additional solutions within the first 65 days. The general solution for the sine function is:

\[ \frac{2\pi}{65} t = \arcsin \left( \frac{13}{21} \right) + 2k\pi \]
or
\[ \frac{2\pi}{65} t = \pi - \arcsin \left( \frac{13}{21} \right) + 2k\pi \]

For the first case:

\[ t = \frac{0.6215 \times 65}{2\pi} + 65k \approx 6.43 + 65k \]

Since \( k = 0 \) fits within 65 days, the solution is \( t \approx 6 \) days (rounded to the nearest day).

For the second case:

\[ t = \frac{(\pi - 0.6215) \times 65}{2\pi} \]

\[ t \approx \frac{(3.1416 - 0.6215) \times 65}{6.2832} \]

\[ t \approx \frac{2.5201 \times 65}{6.2832} \]

\[ t \approx 26.06 \text{ days} \]

Since this also fits within 65 days, the solution is \( t \approx 26 \) days (rounded to the nearest day).

Thus, the times when the brightness is 14 are:

\[ t \approx 6 \text{ days} \]
and
\[ t \approx 26 \text{ days} \]

Would you like more details or have any questions?

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Here are 5 questions you might want to ask:

1. How do you find the period of a trigonometric function?
2. What are the steps to solve a trigonometric equation involving sine?
3. Can you explain how the general solution for sine functions is determined?
4. How can you check for additional solutions within a specific range?
5. What are some applications of trigonometric modeling in real-world scenarios?

**Tip:** When solving trigonometric equations, always consider the periodic nature of the functions to find all possible solutions within the given range.

Learn how to solve the trigonometric equation \( B(t) = 7.5 + 10.5 \sin \left( \frac{2\pi}{65} t \right) \) to find the time when brightness equals 14. Detailed step-by-step solution including the arcsine function and periodicity of sine functions. Suitable for advanced high school students and beyond.

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