Question

A sector is formed if a part of a circle is cut off from the center of a circle. It can then be used to form a cone if you glue the two radii together. For any given sector, the cone thus formed is unique. This means that both the surface area and the volume can be expressed in terms of variable theta (in radian) and R.

Based on the diagrams, what is the relationship between R and L. The answer is R = L.

The question after is express r in terms of theta and R. I've put r = R / theta but I'm sure that's wrong.

Answer 1

http://www.mathwarehouse.com/solid-geometry/cone/images/volume-cone-formula-vs-cylinder-base.png

Answer 2

\(\Huge\bf \color{green}{Welcome~to~OpenStudy!!}\hspace{-310pt}\color{cyan}{Welcome~to~OpenStudy!!}\hspace{-307.1pt}\color{purple}{Welcome~to~\color{purple}{Open}}\color{purple}{Study!!!!}\)
Intense, do you still need help?

Answer 3

seriously, how do you do that kind of text?

Answer 4

yes but it's for a different question. it says express r in terms of theta and R. It's based off of the picture i uploaded. I put r = R/theta

Answer 5

It's called LaTeX in the LaTeX section:) You just gotta learn the codes and that kinda stuff.

Answer 6

Can you take a screenshot? I feel like something is missing.

Answer 7

Those two picture are the only ones on the sheet. Here's what the message says: A sector is formed if a part of a circle is cut off from the center of a circle. It can then be used to form a cone if you glue the two radii together. For any given sector, the cone thus formed is unique. This means that both the surface area and the volume can be expressed in terms of variable theta (in radian) and R. Based on the diagrams, what is the relationship between R and L. The answer is R = L. The question after is express r in terms of theta and R. I've put r = R / theta but I'm sure that's wrong.

Answer 8

What is the length of the sector of radius R and central angle \(\theta\)?

Answer 9

There was no length given. They just want a general equation for r

Answer 10

The arc length of the sector of radius R and central angle \(\theta\) is R\(\theta\) where \(\theta\) is in radians. This arc length becomes the circumference of the circle that forms the base of the cone. Therefore, \(2\pi r = R\theta\). \(\Large r = \frac{R\theta}{2\pi}\)

Answer 11

:D Thanks! The next questions require me to find area of the sector, lateral area of the cone, and the height of the cone. I would just have to substitute \[r = R \theta / 2\pi\]

Answer 12

@aum for finding the area of a sector, could i substitute \[r=R ^{2}\theta ^{2} \div 2\] for \[\pi \times r^{2}\]? I got \[A=R^2\theta^2/4\pi \] Would that be right?

Answer 13

I mean \[r = R^2\theta^2 / 2\pi\]

Answer 14

Area of the sector = \(\large \frac 12 R^2 \theta\).

Answer 15

Lateral Surface Area of the cone = \(\large \pi r L = \pi * \frac{R\theta}{2\pi} * R = \frac 12 R^2 \theta\)

Answer 16

How did you get 1/2 for the area?

Answer 17

Which one: sector or lateral surface area ?

Answer 18

Sector. Did you substitute it into 2 * pi * r?

Answer 19

That is the standard formula for area of a sector: \(\large \frac 12 R^2 \theta\)

Answer 20

Ok thanks and for the lateral surface area of the cone, 1/2 *R^2* theta is the result when everything has been simplified? It is the same as the area, right?

Answer 21

correct. The standard formula for the lateral surface area of a cone is pi * r * L.
I substituted r in terms R and L = R, simplified it and it came out to be the same as the area of the sector as expected.

Answer 22

Area of a sector: http://en.wikipedia.org/wiki/Circular_sector

Answer 23

:D THANK YOU AUM!

Answer 24

You are welcome.