I can see the math problems in your image. Let me type out and solve them step by step for you.

1. Distance from Earth:

The problem states:

The Moon is about 3500 kilometers in diameter. If you could travel around the Moon's equator, how far would you travel? (Use the formula for the circumference of a circle.)

Solution: The formula for the circumference of a circle is: $C = \pi \times d$ Where:

• $\pi \approx 3.14159$
• $d = 3500 \, \text{km}$ (the Moon's diameter)

$C = 3.14159 \times 3500 = 10995.6 \, \text{km}$

So, the circumference of the Moon is approximately 10,995.6 km.

2. Days in a Month:

The problem states:

The Moon takes approximately 29.5 days to complete its orbit around the Earth. If you start counting from a New Moon, how many days until the next New Moon?

Solution: Since one full lunar cycle (New Moon to New Moon) takes 29.5 days, the time until the next New Moon is exactly 29.5 days.

3. Phases of the Moon:

The problem states:

If the Moon goes through 8 distinct phases in one cycle, how many phases does it complete in 3 cycles?

Solution: Each lunar cycle has 8 phases. In 3 cycles, the number of phases is: $8 \times 3 = 24$

So, the Moon completes 24 phases in 3 cycles.

4. Full Moon Visibility:

The problem states:

If the Full Moon rises at sunset and sets at sunrise, how many hours is it visible in the sky if sunset is at 6 PM and sunrise is at 6 AM?

Solution: The Full Moon is visible from sunset (6 PM) to sunrise (6 AM). The total time is: $6 \, \text{PM} \to 6 \, \text{AM} = 12 \, \text{hours}$

So, the Full Moon is visible for 12 hours.

5. Gibbous Moon:

The problem states:

A waxing gibbous Moon is visible for 10 hours before it sets. If it rises at 3 PM, what time does it set?

Solution: If the Moon rises at 3 PM and is visible for 10 hours, the setting time is: $3 \, \text{PM} + 10 \, \text{hours} = 1 \, \text{AM}$

So, the waxing gibbous Moon sets at 1 AM.

6. Earthshine:

The problem states:

If the Earth reflects sunlight onto the Moon, making the unlit side visible, and the Earth is 50 times brighter than a full Moon, how many times brighter is the Earth compared to a crescent Moon that is only $\frac{1}{4}$ lit?

Solution: If the crescent Moon is $\frac{1}{4}$ as bright as a full Moon, the Earth is 50 times brighter than a full Moon. Thus, the Earth is: $50 \div \frac{1}{4} = 50 \times 4 = 200$

So, the Earth is 200 times brighter than the crescent Moon.

Do you have any questions about these solutions? Let me know if you'd like further clarification on any of the steps.

Related Questions to Explore:

1. What is the average orbital speed of the Moon around the Earth in kilometers per hour?
2. How does the Moon's gravitational pull affect the Earth's tides?
3. What are the characteristics of a waxing crescent Moon, and how is it different from a waning crescent Moon?
4. How does the distance between the Earth and Moon vary during a lunar month?
5. Why does the same side of the Moon always face the Earth (tidal locking)?

Math Tip: When solving problems involving the Moon's orbit or phases, always pay attention to the time units (hours, days, etc.) and whether you're working with fractions of a cycle or full cycles.