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# A 288 degree circular sector

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A 288 degree circular sector with radius 18 is rolled to form a cone. Find the height of the cone.

Nov 15, 2020

#1
+9198
+2

We can draw a right triangle where the hypotenuse is the slant height and the legs are the radius and the height of the cone. Then if we can find the slant height and the radius of the base of the cone, we can use the Pythagorean theorem to find the height.

The slant height of the cone is the same as the radius of the sector, which is 18

Now we need to find the radius of the base of the cone.

The circumference of the base is the same as the length of the sector, which we can find like this:

$$\frac{\text{arc length of sector}}{\text{circumference of circle}}=\frac{288^\circ}{360^\circ} \\~\\ \frac{\text{arc length of sector}}{2\pi(\text{radius of circle})}=\frac{288^\circ}{360^\circ} \\~\\ \frac{\text{arc length of sector}}{2\pi(18)}=\frac{288^\circ}{360^\circ} \\~\\ \text{arc length of sector}=\frac{288^\circ}{360^\circ} \cdot 2\pi(18)\\~\\ \text{arc length of sector}=28.8\pi$$   (The circle here is the circle from which the sector is taken)

And so

circumference of base   =   28.8 pi

radius of base   =   circumference of base  /  (2 pi)   =   28.8 pi / (2 pi)   =   14.4

Now that we know the radius of the base of the cone and we know the slant height of the cone, we can use the Pythagorean theorem to find the height of the cone.

(radius of base)2 + (height)2   =   (slant height)2

Plug in what we know...

14.42  +  ( height )2   =   182

Subtract  14.42  from both sides

( height )2   =   182 - 14.42

Simplify the right side

( height )2   =   116.64

Take the positive square root of both sides

height   =   10.8

Nov 15, 2020

#1
+9198
+2

We can draw a right triangle where the hypotenuse is the slant height and the legs are the radius and the height of the cone. Then if we can find the slant height and the radius of the base of the cone, we can use the Pythagorean theorem to find the height.

The slant height of the cone is the same as the radius of the sector, which is 18

Now we need to find the radius of the base of the cone.

The circumference of the base is the same as the length of the sector, which we can find like this:

$$\frac{\text{arc length of sector}}{\text{circumference of circle}}=\frac{288^\circ}{360^\circ} \\~\\ \frac{\text{arc length of sector}}{2\pi(\text{radius of circle})}=\frac{288^\circ}{360^\circ} \\~\\ \frac{\text{arc length of sector}}{2\pi(18)}=\frac{288^\circ}{360^\circ} \\~\\ \text{arc length of sector}=\frac{288^\circ}{360^\circ} \cdot 2\pi(18)\\~\\ \text{arc length of sector}=28.8\pi$$   (The circle here is the circle from which the sector is taken)

And so

circumference of base   =   28.8 pi

radius of base   =   circumference of base  /  (2 pi)   =   28.8 pi / (2 pi)   =   14.4

Now that we know the radius of the base of the cone and we know the slant height of the cone, we can use the Pythagorean theorem to find the height of the cone.

(radius of base)2 + (height)2   =   (slant height)2

Plug in what we know...

14.42  +  ( height )2   =   182

Subtract  14.42  from both sides

( height )2   =   182 - 14.42

Simplify the right side

( height )2   =   116.64

Take the positive square root of both sides

height   =   10.8

hectictar Nov 15, 2020
#2
+114221
+1

Nice, hectictar  !!!!

CPhill  Nov 15, 2020
#3
+846
+2

A 288-degree circular sector with a radius of 18 is rolled to form a cone. Find the height of the cone.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Shortcut:

The radius of a sector:      R = 18

The radius of a cone:      r = R * (288 / 360) = 14.4

Cone slant height:          L = 18

Cone height:                  h = sqrt(182 - 14.42) = 10.8

Nov 16, 2020
edited by jugoslav  Nov 16, 2020
#4
+114221
0

Very nice, jugoslav  !!!!

CPhill  Nov 16, 2020