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.