Question

Semi circular plate is rolled to form a cone. The angle between the generator and axis of cone is

Answer 1

Consider a semicircular plate having radius r.

Now, the Semi circular plate is rolled to form a cone.

There will be two generators, and one vertex which was the center of the semicircular plate .

The two generators are slant height of cone which will be equal to radius r.

If you consider the angle between two generators ,

The angle between the generator and axis of cone will always be less than 90°.

But , if you want to find the exact angle between ,generator and axis of cone , you have to divide semicircle in shapes resembling to triangles,which you will call cone, such that angle between two generators which is equal to radius will lie between 0° < Angle≤90° and sum of all the angles between the conical shapes will be 180°.

Suppose the Semi circular plate is just folded once from the middle to form a cone.

Perimeter of semicircle =  π r

Let base of cone which is in the shape of circle have radius equal to R.

2 π R= π r

R=  $\frac{r}{2}$

→→(Height of cone)² = (slant height)² + (radius)²

→→(H)²= r² +   $[\frac{r}{2}]^2$

→→H$=\frac{\sqrt{5}r}{2}$

Consider semi vertical angle between ,generator and height of cone to be A,then

Tan A

$\frac{R}{l}\\\\ = \frac{r}{2 r}\\\\ = \frac{1}{2}$

Tan 2 A

$=\frac{2 tan A}{1 +tan^2 A}\\\\ = \frac {2 \times \frac{1}{2}}{1+(\frac{1}{2})^2}\\\\ Tan 2A=\frac{4}{5}\\\\ A=\frac{1}{2}Tan^{-1}(\frac{4}{5})$

This will not be the exact answer. it totally depends on how you have rolled the semicircular plate.

But one thing which is sure for this question is ,  

→→ angle between the generator and axis of cone will always be< 90°