SOLUTION: A sector of a circle 7cm subtending an angle 210 at the centre of the circle, is used to form a cone. calculate to the nearest whole number, the base radius of a cone?, height of t

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Question 1037863: A sector of a circle 7cm subtending an angle 210 at the centre of the circle, is used to form a cone. calculate to the nearest whole number, the base radius of a cone?, height of the cone?, total surface area of a cone?, volume of the cone and vertical angle of the cone
Found 2 solutions by josgarithmetic, KMST:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
The 7 cm sector is from part of a circumference, so 7%2Fc=210%2F360 to get to the full circumference if it were.

c%2F7=36%2F21
c%2F7=3%2A18%2F%283%2A7%29
c=7%2818%2F7%29
c=18
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What is the radius for this sector?
If x is the radius, then based on the completed circle,
2pi%2Ax=18
x=18%2F%282pi%29
highlight%28x=9%2Fpi%29--------the sector's radius*.


Once the unattached sector sides are attached, the circumference of the cone becomes the 7 cm length, so 2pi%2Ar=7;
highlight%28r=7%2F%282pi%29%29-----base radius of the cone.


* What was the sector's radius becomes the slant height of the cone when formed. Imagine a cross section right triangle formed in making the cone. The cone height will be h%5E2%2B%287%2F%282pi%29%29%5E2=%289%2Fpi%29%5E2.
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h%5E2=%2881%2Fpi%5E2%29-%2849%2F%284%28pi%29%5E2%29%29
h%5E2=%281%2Fpi%29%5E2%2881-49%2F4%29
h%5E2=%281%2Fpi%29%5E2%28324%2F4-49%2F4%29
h%5E2=%281%2Fpi%29%5E2%28275%2F4%29
h=sqrt%28275%29%2F%282pi%29
highlight%28h=5%2Asqrt%2811%29%2F%282pi%29%29------How tall the cone


You can do the rest.

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Is the 7cm measure the radius of the circle? Or is it the diameter?

Assuming the radius is 7cm,
that would be the slant height of the cone

The circumference of the circle measures
2%2Api%2A%287cm%29=14picm .
The curved edge of the sector measures 210%2F360=7%2F12 as much as the total circumference, or
%287%2F12%2914picm=49pi%2F6cm ,
and that length becomes the circumference of the base of a cone of radius r , so
2pi%2Ar=49pi%2F6cm .
2pi%2Ar=49pi%2F6cm ---> r=49pi%2F12pi=49%2F12cm=approximately4.08cm=approximatelyhighlight%284cm%29 (rounded to the nearest whole number).

A cross section of the cone looks like this triangle:
.
The altitude of length green%28h%29 divides the triangle into two right triangles.
In each of them the legs measure green%28h%29 and r ,
and the hypotenuse mesures 7cm .
Using the Pythagorean theorem, and the approximate value r=4cm ,
we can calculate green%28h%29 :
green%28h%29%5E2%2B%284cm%29%5E2=%287cm%29%5E2
green%28h%29%5E2%2B16cm%5E2=49cm%5E2
green%28h%29%5E2=49cm%5E2-16cm%5E2
green%28h%29%5E2=33cm%5E2
green%28h%29=sqrt%2833%29cm=about5.74cm
So, The height of the cone, to the nearest whole cm is highlight%286cm%29 .

The surface area of the a whole circle of radius 7cm is
pi%2A%287cm%29%5E2=49picm%5E2 .
The surface area of the sector is 210%2F360=7%2F12 as much as the surface area of the a whole circle, or
%287%2F12%29%2A49picm%5E2=343pi%2F12cm%5E2=about89.79cm%5E2 .
(If you use the approximation pi=22%2F7 ,
the area of the a whole circle is calculated as pi%2A%287cm%29%5E2=%2822%2F7%2949cm%5E2=154cm%5E2 ,
and the area of the sector is calculated as %287%2F12%29%28154cm%5E2%29=%28539%2F6%29cm%5E2 ,
which is about540%2F6cm%5E2=90cm%5E2 ).
The surface of the sector becomes the lateral surface of the cone,
so the lateral surface area of the cone is 90cm%5E2 (to the nearest whole number).

It is not clear from the problem is what is asked for is the total surface area of a cone with those dimensions, or just the lateral area.
The base of the cone, which was not formed out of the sector of the circle,
is a circle of radius r=approximately4cm as found above.
The total surface area of a cone with those dimensions is the lateral surface plus the area of the base.
The area of a circle of radius 4cm is
pi%2A%284cm%29%5E2=pi%2A16cm%5E2=about50cm%5E2 (to the nearest whole number),
so the total surface area for a cone like the one in the problem would be
90cm%5E2%2B40cm%5E2=130cm%5E2 .

The volume of a cone with base radius r=4cm and height h=6cm is
%281%2F3%29%2Api%2A%284cm%29%5E2%2A%286cm%29=approximatelyhighlight%28101cm%5E3%29 (to the nearest whole number).

The vertical angle can be calculated from the cross section measurements>
. sin%28red%28A%29%29=4cm%2F%227+cm%22=about0.60825 (rounded).
From that we get red%28A%29=approximately34.85%5Eo ,
and the vertical angle is 2red%28A%29=about2%2A34.85%5Eo=about69.7%5Eo .
So the vertical angle of the cone (in whole degrees is highlight%2870%5Eo%29 .