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Let's solve the first problem (Problem 46) step by step.

### Problem 46:
We are given a sphere with a center \( O \) and radius \( 13 \). A plane intersects the sphere, and the center of the resulting circle is at \( R \), which is at a distance of 5 units from \( O \). We are asked to determine the area of the circle formed by the intersection.

### Step 1: Understanding the Geometry
- The radius of the sphere is \( 13 \).
- The distance between the center of the sphere \( O \) and the center of the circle \( R \) is \( 5 \).

To find the radius of the circle formed by the intersection, we can apply the Pythagorean theorem because the radius of the sphere, the radius of the circle formed by the intersection, and the distance between the centers form a right triangle.

### Step 2: Applying the Pythagorean Theorem
Let \( r \) be the radius of the circle formed by the intersection. According to the Pythagorean theorem:
\[
r^2 + 5^2 = 13^2
\]
\[
r^2 + 25 = 169
\]
\[
r^2 = 169 - 25 = 144
\]
\[
r = \sqrt{144} = 12
\]

### Step 3: Calculating the Area of the Circle
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
Substitute \( r = 12 \):
\[
A = \pi (12^2) = 144\pi
\]

Thus, the area of the circle is \( 144\pi \). Since the problem asks for the area without involving \( \pi \), the correct answer is **144**.

### Answer: 
The correct option is **C. 144**.

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Would you like me to go over the next part of the problem or clarify anything here? Let me know! 

Here are five related questions to extend your understanding:
1. How would the area change if the distance between \( O \) and \( R \) was different?
2. How can you find the volume of the sphere?
3. What is the general formula for the area of a circle when intersecting a sphere?
4. How would the result change if the sphere had a different radius?
5. Can this concept be extended to ellipses if the plane intersection is not perpendicular?

**Tip:** When solving problems involving 3D geometry, visualizing the geometric configuration and using the Pythagorean theorem often simplifies the solution!

La figura muestra una esfera de centro O y radio 13. Al interceptar un plano con la esfera, se determina la circunferencia de centro R. Si la distancia entre O y R es 5, el área de la circunferencia de centro R es:

Learn how to calculate the area of a circle formed by the intersection of a plane with a sphere. Using the Pythagorean theorem and the formula for the area of a circle, this geometry problem involves finding the radius of the circle when the sphere's radius is 13 and the center-to-center distance is 5. Suitable for students in Grades 9-12.

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