2)The radius of a cylinder is 5/6 its height. Find the total surface area of the cylinder if its volume is 150pi.

3)A prism and a cone have the same base area and the same height. The volume of the prism is 1. What is the volume of the cone?

8)In the diagram, XY and XprimeYprime are diameters of the two bases of the cylinder, and XY is parallel to XprimeYprime. We know [XYYprimeXprime] = 30. Find the lateral surface area of the cylinder.

Picture: https://latex.artofproblemsolving.com/5/d/5/5d52ed72b984cb05eb5c13af7b73506d964ac628.png

9)A 288 degree circular sector with radius 15 is rolled to form a cone. Find the height of the cone.

10)The two bases of a right conical frustum have radii 12 and 9. The two bases are 4 units apart. Let the volume of the frustum be V cubic units and the total surface area of the frustum be A square units. Find V + A.

11)The volume of regular octahedron PABCDQ is 243. Rotating octahedron PABCDQ about its diagonal PQ the path of the octahedron forms a new 3D shape. What is the volume of this 3D shape?

Picture: https://latex.artofproblemsolving.com/e/c/1/ec166c676365a480c3be6bb4646ec8c18068c507.png

FiestyGeco
Apr 11, 2018

#1**+1 **

2)The radius of a cylinder is 5/6 its height. Find the total surface area of the cylinder if its volume is 150pi.

We have

150 pi = pi [ 5/6 h ] ^2 * h divide out pi

150 = (25/36) h^3 multiply both sides by 36/25

150 (36/25) = h^3

216 = h^3

6 = h

So...r = (5/6)h = (5/6)(6) = 5

So....the surface area is

2pi * r [ r + h ] =

2pi * (5) [ 5 + 6 ] =

2pi * 55

110 pi units^2

CPhill
Apr 11, 2018

#2**+1 **

9)A 288 degree circular sector with radius 15 is rolled to form a cone. Find the height of the cone.

The perimeter of the base of the cone will be :

2 * pi * 15 (288/360) =

pi * 30/ 360 * 288

pi * 288/12 =

24 pi

So....the radius of the cone is given by

24pi = 2pi * r divide both sides by 2 pi

12 = r

The slant height will be the original radius of the sector = 15

And this forms the hypotenuse of a 9 - 12 - 15 right triangle

And the radius forms one of the legs = 12

So....the height forms the other leg = 9

CPhill
Apr 11, 2018

#3**+1 **

10)The two bases of a right conical frustum have radii 12 and 9. The two bases are 4 units apart. Let the volume of the frustum be V cubic units and the total surface area of the frustum be A square units. Find V + A.

The height = 4

The volume, V, is given by

(1/3) * pi * height ( [radius base 1]^2 + product of the base radii +[ radius base 2]^2 ) =

pi / 3 * 4 * ( 12^2 + 12*9 + 9^2)

pi / 3 * 4 * ( 144 + 108 + 81 )

pi / 3 * 4 * ( 333)

pi * 4 * 111

444 pi

The surface area A, is given by :

pi ( sum of radii) * √ [ (difference of radii)^2 + height^2 ] + pi (radius 1)^2 + pi (radius 2)^2

pi ( 21) √ [ 3^2 + 4^2 ] + pi [ 144 + 81 ]

pi [ 21 * 5 + 225 ]

pi [ 105 + 225 ]

pi [ 330]

330 pi

So

V + A =

[444 +330 ] pi =

774 pi

CPhill
Apr 11, 2018

#4**+1 **

3)A prism and a cone have the same base area and the same height. The volume of the prism is 1. What is the volume of the cone?

The base area of the cone = pi * r^2

But...this is also the base area of the prism

So.....the volume of the prism = base area * height = 1

So

1 = base area * height

1 = pi (r^2) * height

1/ [ pi (r^2)] = height

So....the volume of the cone is

(1/3)[ pi * r^2] [ 1 / [ pi * ^2 ] = 1/3

CPhill
Apr 11, 2018

#5**+1 **

8)In the diagram, XY and XprimeYprime are diameters of the two bases of the cylinder, and XY is parallel to XprimeYprime. We know [XYYprimeXprime] = 30. Find the lateral surface area of the cylinder.

I'm assuming that [ XYY'X'] represents the* area *of the cross-section....then the area can also be represented by

2r * h

So

2r * h = 30

r * h = 15

So....the lateral surface area, based on this assumption is

2pi *r * h =

2pi (15) =

30 pi units^2

CPhill
Apr 12, 2018