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The problem involves a cone with a base radius of 1 and height \(\sqrt{3}\), and a plane \(p\) intersecting the cone at an angle of \(30^\circ\) with the horizontal. We need to find the area of the cross section formed when the plane passes through the cone.

### Step 1: Understand the geometry
- The base of the cone is a circle with radius 1, lying on the horizontal plane.
- The height of the cone is \(\sqrt{3}\), and it reaches its apex directly above the center of the base.
- A plane \(p\) intersects the cone, forming a \(30^\circ\) angle with the horizontal. 

### Step 2: Determine the shape of the cross-section
When a plane intersects a cone, the cross-section is generally an ellipse. If the plane is parallel to the base, the cross-section would be a circle, but since the plane intersects at an angle, the cross-section will be an ellipse.

### Step 3: Find the dimensions of the ellipse
For a right circular cone with height \(h\) and base radius \(r\), if a plane intersects the cone at an angle \(\theta\) with the horizontal, the semi-major axis \(a\) and semi-minor axis \(b\) of the resulting ellipse can be given by:
\[
a = \frac{r}{\cos(\theta)} \quad \text{and} \quad b = r
\]
Here \(r = 1\), \(h = \sqrt{3}\), and \(\theta = 30^\circ\). So:
\[
a = \frac{1}{\cos(30^\circ)} = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}}
\]
\[
b = r = 1
\]

### Step 4: Calculate the area of the ellipse
The area \(A\) of an ellipse is given by:
\[
A = \pi \times a \times b = \pi \times \frac{2}{\sqrt{3}} \times 1 = \frac{2\pi}{\sqrt{3}}
\]

### Step 5: Simplify and express the area
We can simplify the area further by rationalizing the denominator:
\[
A = \frac{2\pi}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\pi \sqrt{3}}{3}
\]
This area is given as \(N\pi\), where \(N^2 = \frac{2\sqrt{3}}{3}\). Thus,
\[
N^2 = \frac{2\sqrt{3}}{3} \Rightarrow N^2 = \frac{m}{n} \Rightarrow m = 6\sqrt{3}, n = 3
\]
However, simplifying further:
\[
\frac{m}{n} = 3N \Rightarrow N = \sqrt{\frac{2}{3}} \Rightarrow m+n= 5
]

The final answer is \( m+n = 5 \)

### Further Questions:
1. What is the general equation of a plane intersecting a cone?
2. How would the solution change if the plane intersected at a different angle?
3. Can the area of the cross-section ever exceed the area of the base?
4. What are other possible shapes for cross-sections of a cone with different planes?
5. How do the values of \(m\) and \(n\) change with different cone dimensions?

### Tip:
Understanding how different geometric shapes interact with planes is key to solving many problems in geometry.

Explore how to calculate the area of the cross-section formed by a plane intersecting a cone at a 30-degree angle. Understand the geometry of cones, ellipses, and the specific angle of intersection. Suitable for advanced high school students and above.

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