The problem asks to calculate the surface area of a Dyson Sphere with a radius of $1.5 \times 10^8$ km. The answer should be in scientific notation. Problem 19 asks how to use the solution of problem 18 and problem 15 to calculate how many times the surface area of the earth the Dyson Sphere is. Problem 20 asks how many planets similar to Earth can be created if the Dyson Sphere material is used to make planets with the same surface area as Earth.

1. Problem Description

The problem asks to calculate the surface area of a Dyson Sphere with a radius of $1.5 \times 10^8$ km. The answer should be in scientific notation. Problem 19 asks how to use the solution of problem 18 and problem 15 to calculate how many times the surface area of the earth the Dyson Sphere is. Problem 20 asks how many planets similar to Earth can be created if the Dyson Sphere material is used to make planets with the same surface area as Earth.

2. Solution Steps

Problem 18:

The formula for the surface area of a sphere is $4 \pi r^2$. Given the radius $r = 1.5 \times 10^8$ km, we can calculate the surface area of the Dyson Sphere.

$A = 4 \pi (1.5 \times 10^8)^2$

$A = 4 \pi (2.25 \times 10^{16})$

$A = 4 \times 3.14159 \times 2.25 \times 10^{16}$

$A = 12.56636 \times 2.25 \times 10^{16}$

$A = 28.27431 \times 10^{16}$

$A = 2.827431 \times 10^{17} \text{ km}^2$

$A \approx 2.83 \times 10^{17} \text{ km}^2$

Problem 19:

Let $A_{Dyson}$ be the surface area of the Dyson Sphere and $A_{Earth}$ be the surface area of the Earth. We found $A_{Dyson}$ in problem

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8. Assume the answer to problem 15 is the surface area of the Earth, $A_{Earth}$. To find how many times the surface area of the Earth the Dyson Sphere is, we divide the surface area of the Dyson Sphere by the surface area of the Earth.

$\text{Number of times} = \frac{A_{Dyson}}{A_{Earth}}$

Without the answer to problem 15, we cannot calculate the numerical answer. However, the method is to divide the Dyson Sphere's surface area by the Earth's surface area.

Problem 20:

The question asks how many planets similar to Earth can have their surface area if the Dyson sphere is completely converted to make planets of the same surface area of Earth. We can divide the surface area of the Dyson sphere by the surface area of the Earth to find the number of Earth-like planets.

$\text{Number of planets} = \frac{A_{Dyson}}{A_{Earth}}$

Again, without the Earth's surface area we can only write an expression but not compute a numerical result.

3. Final Answer

Problem 18: $2.83 \times 10^{17} \text{ km}^2$

Problem 19: $\frac{A_{Dyson}}{A_{Earth}}$, where $A_{Dyson}$ is the surface area of the Dyson Sphere ($2.83 \times 10^{17} \text{ km}^2$) and $A_{Earth}$ is the answer to problem 15 (surface area of Earth).

Problem 20: $\frac{A_{Dyson}}{A_{Earth}}$, where $A_{Dyson}$ is the surface area of the Dyson Sphere ($2.83 \times 10^{17} \text{ km}^2$) and $A_{Earth}$ is the surface area of Earth.