1. Rubber bungs are made by removing the tops of cones. Starting
with a cone of radius 10cm and height 16cm, a rubber bungs made 
by cutting a cone of radius 5 cm and height 8cm from the top. 
Find the volume and total surface area of the rubber bung.

They start with a rubber cone whose cross section is like this:


             /|\
            / | \
         S /  |  \
          /   |16 \
         /    |    \
        /_____|_____\
           10

This has a volume of pr³/3 = p(10)³/3 = 1000p/3

Its surface area is found by
                          
Lateral Surface Area = prS, where S = slant height

(Note: Lateral Surface Area does not include the circular base)

Lateral Surface Area = prS = p(10)S = 10pS 
     _______    ___________    _______    ___    _____     __ 
S = Ör² + h² = Ö(10)²+(15)² = Ö100+225 = Ö325 = Ö25·13 = 5Ö13 
                                   __        __
Lateral Surface Area = prS = 10p(5Ö13) = 50pÖ13

Base (circle) Surface Area = pr² = p(10)² = 100p

Total Surface Area = Lateral surface area + Base surface area  
                         __               __
Total Surface Area = 50pÖ13 + 100p = 50p(Ö13 + 2)

------------------------------------------------------

Then they cut away a rubber cone whose cross section is like this:

             /|\
          S / |8\
           /__|__\
            5
This has a volume of pr³/3 = p(5)³/3 = 125p/3

Its surface area is found by
                          
Lateral Surface Area = prS, where S = slant height

Lateral Surface Area = prS = p(5)S = 5pS 
     _______    _________    _____    __    
S = Ör² + h² = Ö(5)²+(8)² = Ö25+64 = Ö89  
                                 __       __
Lateral Surface Area = prS = 5p(Ö89) = 5pÖ89

Note: They cut away only the lateral surface area of this small cone.
(Its base circle must then be added to find the total surface 
area of the bung)   

-----------------------------------------

They end up with a rubber bung whose cross section is like this:

           _5_____
          /   |   \
         /    |    \
        /_____|_____\
          10

Volume of bung = Volume of large cone - Volume of small cone  

Volume of bung = 1000p/3 - 125p/3 = 875p/3

That's approximately 916.3 cubic centimeters.

Total Surface Area of bung =
   Total Surface Area of large Cone - 
        Lateral Surface Area only of small Cone +
             Area of base of small cone (circular top of bung)

Area of circular top of bung = pr² = p(5)² = 25p
                                  __           __
Total Surface Area of bung = 50p(Ö13 + 2) - 5pÖ89 + 25p 
                                    __         __             
Total Surface Area of bung = 5p[10(Ö13 + 2) - Ö89 + 5]
                                   __         __             
Total Surface Area of bung = 5p[10Ö13 + 20 - Ö89 + 5]
                                   __    __             
Total Surface Area of bung = 5p(10Ö13 - Ö89 + 25)

That's approximately 810.9 square centimeters.
                                 
-----------------------------------------

Edwin