Determine the height of the Earth's shadow cone, which is illuminated by the sun if the sun has a diameter equal to 109 times
Of the Earth and the distance from Earth to the Sun is about 15 × 10 ^ 7 km.

To determine the height of the Earth's shadow cone (often referred to as the "umbra" length) formed when the Earth blocks sunlight, we can use the geometry of similar triangles. Here's how:

### Step 1: Understanding the Problem
- The Sun and the Earth form similar triangles with the shadow cone.
- The Sun’s diameter (\( D_s \)) is 109 times that of the Earth's diameter (\( D_e \)).
- The distance between the Earth and the Sun (\( d \)) is \( 15 \times 10^7 \) km.

### Step 2: Known Values
- Let \( D_e \) be the Earth's diameter.
- Therefore, \( D_s = 109 \times D_e \).
- Distance between Earth and Sun: \( d = 15 \times 10^7 \) km.

### Step 3: Geometry of Similar Triangles
Using the similar triangles concept, the relationship between the distances and diameters is given by:

\[
\frac{L}{L - d} = \frac{D_s}{D_e}
\]

Where:
- \( L \) is the length of the Earth's shadow cone (height of the shadow).
- \( d \) is the distance from the Earth to the Sun.
- \( D_s \) is the Sun's diameter.
- \( D_e \) is the Earth's diameter.

### Step 4: Substitute Known Values

Substituting the known values:

\[
\frac{L}{L - 15 \times 10^7} = \frac{109 \times D_e}{D_e} = 109
\]

Simplify the equation:

\[
\frac{L}{L - 15 \times 10^7} = 109
\]

Multiplying both sides by \( L - 15 \times 10^7 \):

\[
L = 109 (L - 15 \times 10^7)
\]

Expanding the equation:

\[
L = 109L - 109 \times 15 \times 10^7
\]

### Step 5: Solving for \( L \)

Rearrange the equation:

\[
L - 109L = -109 \times 15 \times 10^7
\]

\[
-108L = -109 \times 15 \times 10^7
\]

\[
L = \frac{109 \times 15 \times 10^7}{108}
\]

Calculating the value:

\[
L \approx 15.14 \times 10^7 \, \text{km}
\]

### Final Answer
The height of the Earth's shadow cone is approximately \( 15.14 \times 10^7 \) km.

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Do you have any questions or need further clarification?

### Related Questions
1. How would the shadow cone length change if the Earth's diameter increased?
2. What is the effect of the Moon's position on the Earth's shadow cone during a lunar eclipse?
3. How does the distance between the Earth and the Sun affect the length of the shadow cone?
4. What happens to the shadow cone during a solar eclipse?
5. How does the size of the Sun's diameter affect the Earth's shadow cone?

### Tip
Always remember that similar triangles are powerful tools in solving geometric problems related to shadows and projections, especially in astronomy.

Determine the height of the Earth's shadow cone, which is illuminated by the sun if the sun has a diameter equal to 109 times of the Earth and the distance from Earth to the Sun is about 15 × 10^7 km.

Learn how to calculate the height of the Earth's shadow cone, given that the Sun's diameter is 109 times that of the Earth, and the distance between the Earth and Sun is 15 × 10^7 km. This problem explores similar triangles and proportionality, ideal for high school students.

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