Question 1183412
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A sector with an angle 110 degrees at the center of a circle is cut away from a circular piece of paper of radius 70cm. 
The remaining part is folded to form a cone. Find 1. the vertical angle of the cone 2. The angle of the sector.


<pre>
After cutting away the sector of 110°, the remaining pert of the circle is the sector of 360° - 110° = 250°.

The length of the arc of the sector of (250°, R = 70 cm) is

    {{{2pi*R*(250/360)}}} = {{{2*3.14159*70*(250/360)}}} = 305.4323611 cm.



The radius of the base of the cone 'r' can be defined from

    {{{2pi*r}}} = 305.4323611,  r = {{{305.4323611/(2*3.14159)}}} = 48.61111... cm.



It is the same as to use equation

    {{{2pi*R*(250/360)}}} = {{{2pi*r}}},  r = {{{R*(250/360)}}} = {{{70*(250/360)}}} = 48.61111... cm.


Now the cone has the slant height of 70 cm and the radius of 48.61111... cm.


From the right-angled triangle,  for the angle 'a' between the slant height and the cone axis we have

    sin(a) = {{{48.61111/70}}} = 0.694444...


Thus angle 'a' is  a = arcsin(0.694444),  or about 44°.


Vertical angle of the cone is about  2*44° = 88°.
</pre>

Solved.