Pls Help, Due Tmrow

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

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

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

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

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°

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