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# some questions

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A, An equilateral triangle of side length 6sqrt3 is rotated about an altitude to form a cone. What is the volume of the cone?

B, A 288 degree circular sector with radius 15 is rolled to form a cone. Find the volume of the cone.

C, What is the radius, in inches, of a right circular cylinder if the lateral surface area is 24 pi square inches and the volume is 24 pi cubic inches?

D, At the MP Donut Hole Factory, Niraek, Theo, and Akshaj are coating spherical donut holes in powdered sugar. Niraek's donut holes have radius 6 mm, Theo's donut holes have radius 8 mm, and Akshaj's donut holes have radius 10 mm. All three workers coat the surface of the donut holes at the same rate and start at the same time. Assuming that the powdered sugar coating has negligible thickness and is distributed equally on all donut holes, how many donut holes will Niraek have covered by the first time all three workers finish their current donut hole at the same time?

(D has been answered before with the solution 144 donut holes but this is not the correct answer unfortunately)

May 16, 2018

#1
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A, An equilateral triangle of side length 6sqrt3 is rotated about an altitude to form a cone. What is the volume of the cone?

The radius of the cone is      6√3 / 2  = 3√3

The height of the cone is   (√3) * 3√3 =   9

So....the volume of the cone  is

(1/3) pi  * radius ^2  * height  =

(1/3) pi  * ( 27) * 9  =

81 pi  units^3   May 16, 2018
#2
+1

B, A 288 degree circular sector with radius 15 is rolled to form a cone. Find the volume of the cone.

The sector radius will from the slant height of the cone.

The  arc length of the sector will form the base circumference of the cone...its length is :

2 pi ( 15) (288/360)   =    24pi   units

So....the radius of the cone is figured as

24 pi  = 2 pi (r)

12  = r

And the height of the cone is

√ [15^2  -12^2 ] = √ [ 225 - 144 ]  = √81  =  9

So....the volume of the cone is

(1/3) pi (radius)^2  * (height)  =

(1/3) pi  * (12)^2  * 9  =

432 pi  units^2   May 16, 2018
#3
+1

C, What is the radius, in inches, of a right circular cylinder if the lateral surface area is 24 pi square inches and the volume is 24 pi cubic inches?

If  the  volume is  24 pi  in^3, we have that

24 pi =  pi ^ r^2  * h

Solving for  h  we have that

24 / r^2  = h       (1)

And if the lateral surface area is  24 pi   in^2......we have that

24 pi  =   2 pi * r * h

Sub (1)  for  h  and we have

24 pi  =   2 pi * r  * (24/r^2 )

1  = 2 / r

r  = 2   in   May 16, 2018
#4
+2

Here's my attempt at D.....

Surface area  of each  donut hole covered by Akshaj  =  4 pi (10)^2  = 400 pi  mm^2

Surface area  of each donut hole  covered by Theo  = 4 pi (8)^2 =  256 pi     mm^2

Surface area of each donut hole covered by Niraek  =  4 pi (6)^2 =  144 pi   mm^2

Ignoring  the  "pi's"   and prime factoring each number we have that

5^2 * 2^4   =  400

2^8  = 256

2^4 * 3^2   =  144

The LCM  of these three numbers is :

2^8 * 5^2 * 3^2   =   57600

So.....when they all finish a donut hole at the same time :

Akshaj  will have covered    57600 / 400   =  144 donut holes

Theo will have covered  57600 / 256  = 225 donut holes

Niraek will have covered  57600 / 144  =  400 donut holes

EDIT to correct an error!!!!   May 16, 2018
edited by CPhill  May 18, 2018