<PRE><B>Question 50803</B>
&lt;pre&gt;&lt;font size = 4&gt;&lt;b&gt;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 &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;r³/3 = &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;(10)³/3 = 1000&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;/3

Its surface area is found by
                          
Lateral Surface Area = &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;rS, where S = slant height

(Note: Lateral Surface Area does not include the circular base)

Lateral Surface Area = &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;rS = &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;(10)S = 10&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;S 
     _______    ___________    _______    ___    _____     __ 
S = &lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;r² + h² = &lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;(10)²+(15)² = &lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;100+225 = &lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;325 = &lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;25·13 = 5&lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;13 
                                   __        __
Lateral Surface Area = &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;rS = 10&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;(5&lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;13) = 50&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;&lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;13

Base (circle) Surface Area = &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;r² = &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;(10)² = 100&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;

Total Surface Area = Lateral surface area + Base surface area  
                         __               __
Total Surface Area = 50&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;&lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;13 + 100&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt; = 50&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;(&lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;13 + 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 &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;r³/3 = &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;(5)³/3 = 125&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;/3

Its surface area is found by
                          
Lateral Surface Area = &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;rS, where S = slant height

Lateral Surface Area = &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;rS = &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;(5)S = 5&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;S 
     _______    _________    _____    __    
S = &lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;r² + h² = &lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;(5)²+(8)² = &lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;25+64 = &lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;89  
                                 __       __
Lateral Surface Area = &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;rS = 5&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;(&lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;89) = 5&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;&lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;89

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  

&lt;font color = &quot;red&quot;&gt;Volume of bung = 1000&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;/3 - 125&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;/3 = 875&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;/3

That&#39;s approximately 916.3 cubic centimeters.&lt;/font&gt;

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 = &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;r² = &lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;(5)² = 25&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;
                                  __           __
Total Surface Area of bung = 50&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;(&lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;13 + 2) - 5&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;&lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;89 + 25&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt; 
                                    __         __             
Total Surface Area of bung = 5&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;[10(&lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;13 + 2) - &lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;89 + 5]
                                   __         __             
Total Surface Area of bung = 5&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;[10&lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;13 + 20 - &lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;89 + 5]
                                   &lt;font color = &quot;red&quot;&gt;__    __             
Total Surface Area of bung = 5&lt;font face=&quot;symbol&quot;&gt;p&lt;/font&gt;(10&lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;13 - &lt;font face=&quot;symbol&quot;&gt;Ö&lt;/font&gt;89 + 25)

That&#39;s approximately 810.9 square centimeters.&lt;/font&gt;
                                 
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Edwin&lt;/pre&gt;&lt;/font&gt;&lt;/b&gt;


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