Answer :
To solve the question, we need to find a relationship between the height (h), the radius (r), the volume (V) and the curved surface area (A) of the cone.
For a cone, the volume (V) is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
The curved surface area (A) is given by the formula:
\[ A = \pi r \sqrt{r^2 + h^2} \]
It is given that the volume and the curved surface area are both 'n'. So we can set up the following equations:
\[ n = \frac{1}{3} \pi r^2 h \]
\[ n = \pi r \sqrt{r^2 + h^2} \]
From the first equation, we can solve for r^2:
\[ r^2 = \frac{3n}{\pi h} \]
Substitute this into the second equation and solve for r:
\[ n = \pi r \sqrt{\frac{3n}{\pi h} + h^2} \]
\[ n = \pi r \sqrt{\frac{3n + \pi h^3}{\pi h}} \]
\[ \frac{n}{\pi r} = \sqrt{\frac{3n + \pi h^3}{\pi h}} \]
Square both sides:
\[ \left(\frac{n}{\pi r}\right)^2 = \frac{3n + \pi h^3}{\pi h} \]
We know that:
\[ r = \sqrt{\frac{3n}{\pi h}} \]
\[ r^2 = \frac{3n}{\pi h} \]
Substitute r^2 into the left side of the equation:
\[ \left(\frac{n}{\pi \sqrt{\frac{3n}{\pi h}}}\right)^2 = \frac{3n + \pi h^3}{\pi h} \]
Simplify the left side:
\[ \left(\frac{n}{\sqrt{3n h/\pi}}\right)^2 = \frac{3n + \pi h^3}{\pi h} \]
Simplify further:
\[ \left(\frac{n}{\sqrt{3n h/\pi}}\right)^2 = \frac{3n + \pi h^3}{\pi h} \]
\[ \frac{n^2}{3n h/\pi} = \frac{3n + \pi h^3}{\pi h} \]
\[ \frac{\pi n^2}{3n h} = \frac{3n + \pi h^3}{\pi h} \]
Now solve for n:
\[ \frac{\pi n^2}{3n} = 3 + \frac{\pi^2 h^2}{3} \]
\[ \frac{\pi n}{3} = 3 + \frac{\pi^2 h^2}{3} \]
\[ n = \frac{9}{\pi} + h^2 \]
We don't need to explicitly find the value of n to find the vertical angle. We now use r in terms of h and n to find the vertical angle.
The vertical angle is given by the angle at the tip of the cone, which can be calculated by:
\[ \theta = 2 \cdot \arctan\left(\frac{r}{h}\right) \]
\[ r = \sqrt{\frac{3n}{\pi h}} \]
\[ \theta = 2 \cdot \arctan\left(\frac{\sqrt{\frac{3n}{\pi h}}}{h}\right) \]
Since we don't have an explicit value for n, we cannot calculate the exact angle. However, if we had the value of n, we could calculate the radius r from the volume formula and then insert it into the arctan function to find the vertical angle.
