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put this solution on YOUR website! 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)
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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.
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Edwin
