The problem you're working on involves a periodic function that describes the proportion of the Moon illuminated each night. The function is given by:

$I(t) = \frac{1}{2} + \frac{1}{2} \cos\left(\frac{2\pi t}{29.5}\right)$ where $t$ is the time in days after January 1st.

Let's go through each part of the question step by step.

Part a:

You are asked to sketch the graph of $I(t)$ against $t$, where $0 \leq t \leq 60$.

This is a cosine function with:

• Amplitude: $\frac{1}{2}$
• Period: $29.5$ days (representing the lunar cycle)
• Vertical shift: $\frac{1}{2}$ (ensuring that the values of $I(t)$ remain between 0 and 1, representing the proportion of the moon illuminated)

Part b:

You are asked to find the proportion of the Moon illuminated on specific dates:

1. January 6th: This corresponds to $t = 6$.
2. January 21st: This corresponds to $t = 21$.
3. January 27th: This corresponds to $t = 27$.
4. February 9th: This corresponds to $t = 40$.

For each of these values of $t$, substitute into the function $I(t)$.

Part c:

You are asked to determine how often a full moon occurs. A full moon occurs when the Moon is fully illuminated, i.e., when $I(t) = 1$.

Since $I(t) = 1$ occurs when $\cos\left(\frac{2\pi t}{29.5}\right) = 1$, you can find the values of $t$ that satisfy this condition.

Part d:

You are asked to determine on what dates between January and February the Moon is not illuminated at all (i.e., when $I(t) = 0$).

This occurs when $\cos\left(\frac{2\pi t}{29.5}\right) = -1$. You can solve for $t$ to find when this happens.

Would you like me to calculate the exact values for part b, c, and d? Let me know if you'd like further details on any of these steps!

Here are 5 related questions you could consider:

1. What is the derivative of $I(t)$, and what does it represent in terms of the lunar cycle?
2. How would the function $I(t)$ change if the lunar cycle period were longer or shorter?
3. Can you explain the significance of the amplitude and vertical shift in the context of this problem?
4. How would you determine the time between two consecutive new moons (i.e., when the Moon is not illuminated)?
5. What are the general characteristics of trigonometric functions, and how do they relate to periodic phenomena like lunar cycles?

Tip: When analyzing periodic functions, always check the period, amplitude, and phase shift to understand their behavior better.