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1)Find the volume of a right circular cone that has radius 8 and slant height 10.

Picture: https://latex.artofproblemsolving.com/0/5/8/058f16e2184d0046cde980764e57cac68add320a.png

 

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?

 

4)The volume of a cylinder is 100. Another cylinder has twice the height but half the base radius. What is the volume of the second cylinder?

 

5)Find the volume of a cone that has 5 as its base radius and a lateral surface area of 65pi.

 

6)The slant height and radius of a cone are 4 and 1, respectively. Unrolling the curved surface gives a circular sector with center angle n degrees. Find n.

 

7)The lateral surface area of a cylinder is 30% of its surface area. What is the ratio between the base radius and the height of the cylinder?

 

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 10, 2018
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6+0 Answers

 #1
avatar+85757 
+1

1.  The height will be the leg  of a 6 - 8 -10 right triangle....so...the height  = 6

 

So...the volume is  (1/3) pi (8)^2 (6)  = 128 pi   units^3

 

 

cool cool cool

CPhill  Apr 10, 2018
 #2
avatar+85757 
+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.

 

If the radius  is 5/6 the height....then the height  is   6/5  the radius

 

So....we can find the radius thusly :

 

150 pi  =  pi (r)^2 (6/5)r

150  = (6/5)r^3   multiply both sides by 5/6

125  = r^3    take the cube root of both sides

5  = r

 

So...the surface area is given by

 

2 pi *r * h  =

2 pi  *(5) * [ (6/5) * 5 ] =

2 pi * (5) (6)

60 pi units^2

 

 

cool cool cool

CPhill  Apr 10, 2018
 #3
avatar+85757 
+1

4)The volume of a cylinder is 100. Another cylinder has twice the height but half the base radius. What is the volume of the second cylinder?

 

100  = pi (r^2) h

Solving for the height, we have

h  = 100/ (pi *r^2)

So....twice the height  =  200 / (pi * r^2)

 

So....the volume, V, of the second cylinder  can be expressed as :

 

V  =  pi (r/2)^2* (200) / (pi * r^2)  =  (pi/pi) (r^2/r^2) (200/ 4)   =  (1)(1)*50  = 50

 

 

cool cool cool

CPhill  Apr 11, 2018
edited by CPhill  Apr 11, 2018
 #4
avatar+85757 
+1

5)Find the volume of a cone that has 5 as its base radius and a lateral surface area of 65pi.

 

The lateral surface area, A,  is given by   

 

A  = pi * radius * slant height

 

65 pi  = pi * 5 * slant height       divide both sides by 5 pi

 

13  = slant height

 

We can use the Pythagorean Theorem to find the height of the cone

 

height  =  √ [ slant height^2  - radius^2 ]  = √[13^2 - 5^2]  = √ [144]  = 12

 

So....the volume of the cone is

 

(1/3) pi (radius^2) (height )  =  (1/3) pi * 5^2 * 12  =  

 

(1/3) pi * 300  =

 

100 pi  units^3

 

 

cool cool cool

CPhill  Apr 11, 2018
 #5
avatar+85757 
+1

6)The slant height and radius of a cone are 4 and 1, respectively. Unrolling the curved surface gives a circular sector with center angle n degrees. Find n.

 

If the radius of the cone is 1, the perimeter of the circular sector  is  2pi (radius) =

2pi (1)  =    2pi

 

So........when we unroll the curved surface, we will have a circular sector with a perimeter of 2pi  and a radius  of 4

 

So....we can find the central angle [in radians] thusly :

 

2pi  = radius * theta in radians

2pi  = 4 * theta in radians      divide both sides by 4

pi/2  = theta in radians

 

And.....pi/2 in radians =  90°

 

 

cool cool cool

CPhill  Apr 11, 2018
 #6
avatar+85757 
+1

7)The lateral surface area of a cylinder is 30% of its surface area. What is the ratio between the base radius and the height of the cylinder

 

Let S be the surface area....then....

S  =   2pi * r  ( r + h)     (1)

Lateral surface area  =   2pi * r * h  = .30S    ⇒  2pi * r  =  .30S / h    (2)

 

Sub (2)  into (1)

 

S  =  (.30S/h) (r + h)   ......  divide through by S

 

1 = (.30/h) (r + h)

 

h / .30   =  r + h

 

h / (3/10)  = r + h

 

(10/3) h = r + h

 

(7/3) h  = r

 

r / h  =  7/3

 

 

cool cool cool

CPhill  Apr 11, 2018

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