Let's break down the problem in your image step-by-step:

Part (a)

The first part asks for the length of the Earth's shadow. You are given a scaled figure, with the following details:

• The radius of the Earth (S) = 432,000 miles
• The distance between the Earth and the Sun = 93,000,000 miles
• The radius of the Sun = 400,000 miles

Using the scale factor method:

The scale ratio can be calculated as: $\frac{\text{Radius of Earth}}{\text{Radius of Sun}} = \frac{432,000}{400,000} = \frac{54}{50}$

Next, we can set up a proportion to find the length $x$ (the length of the shadow): $\frac{432,000}{400,000} = \frac{93,000,000}{x}$ Cross multiplying: $432,000 \times x = 400,000 \times 93,000,000$ Solving for $x$: $x = \frac{400,000 \times 93,000,000}{432,000} = 86,111,111.1 \text{ miles}$

Thus, the length of the Earth's shadow is approximately 86,111,111.1 miles.

Part (b)

This part asks you to compare the length of the Earth's shadow to the distance between the Earth and the Moon, which is 240,000 miles.

From part (a), we found that the Earth's shadow is about 86,111,111.1 miles. Since the shadow length (86,111,111.1 miles) is much greater than the distance between the Earth and the Moon (240,000 miles), it suggests that a lunar eclipse can occur.

Conclusion

• The shadow is much longer than the distance from the Earth to the Moon (240,000 miles), meaning the Moon can pass through the Earth's shadow, allowing a lunar eclipse to occur.

Would you like more clarification or details on any of these steps? Feel free to ask!

Here are some follow-up questions you may find useful:

1. What is the significance of the scale factor used in this problem?
2. How would the answer change if the distance from the Earth to the Moon were different?
3. What other factors could influence whether a lunar eclipse occurs?
4. How does the position of the Earth, Moon, and Sun affect the occurrence of a lunar eclipse?
5. What other astronomical phenomena are related to eclipses?

Tip: Always double-check the units and scale when dealing with problems involving proportions and distances to ensure that the calculations align with the real-world measurements!