3D Geometry

1.  The lateral surface area =  pi * r * L     where L is the slant height of the  cone

So

16sqrt (5) * pi = pi * 4 * L

16sqrt (5) / 4 = L  =  4sqrt (5)

Height of cone =  sqrt [ L^2 - r^2]  =  sqrt [ 80 - 4^2] = sqrt [ 64] = 8

Volume of  cone  =     pi * r^2 * height  / 3   =   pi * 4^2 * 8  / 3   =   (128/3) pi 

2.  The height of the triangle is   sqrt [ 5^2 - 4^2 ] = sqrt [9] = 3 

Rotating this triangle in space around BC produces identical  cones with radiuses = 3  and  heights =  4

So.....the volume is

2 * [  pi * radius^2 * height   / 3 ]  =

2 * [  pi * (3)^2  * 4   / 3 ]   =

24 pi

CORRECTED

3. We can find the radius of the circle, r,  as

14pi =  pi*r^2

14  = r^2

r =sqrt (14)

We can find the radius of the sphere, R,  using the Pythagorean Theorem

R = sqrt [ (sqrt (14))^2  + 2^2  ] = sqrt  [ 18] 

Surface area of sphere =

4 pi ^ R^2  =

4 * pi * 18   =

72 pi

4. Volume of a Frustum  =

(1/3) pi * h * ( r^2  + Rr + R^2)

Where  

h = height

r = radius of smaller base

R = radius of larger base

So

(1/3) pi* 4  * (3^2 + 6*3 + 6^2)  =

(4/3) ( 9 + 18 + 36) pi  =

(4/3) (63)  pi  =

84 pi

(4) 

Surface area

SA  = pi  * L *  ( R + r)  +   pi ( R^2 + r^2)

Where L  = the slant height  = sqrt [ H^2  +  (R - r)^2 ]  = sqrt [ 4^2  + (6-3)^2 ] = sqrt [ 16 + 9 ] =  5

So

SA =  pi * 5 (6 + 3) + pi  ( 6^2 + 3^2)  = 

pi (45)  + pi ( 45)  = 

90 pi