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# 2015 NS 30

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304
2 Answer: 0.343

Mar 16, 2019

#1
+1

See the following image : Triangle AJB represents the cross-section of the cone

It's volume is  (1/3)(pi)(5^2)(12)  =  100 pi cm^3

We are trying to find a similar cross-section  that will include all the points inside the cone that will be at least 1cm  from the surface area of the cone....call this similar cross-section KGH

We need to find point "H".....this will have to be 1 unit from the cross-section diameter JB of the cone and 1 unit from the right side of the cone's cross-section which can be represented by the line y = (-12/5) ( x - 5)

In standard form the line is given by :   12x + 5y - 60   = 0

We know that  the y coordinate of H = 1

So....using the formula for the distance from a point to a line, we can find the x coordinate of "H"....note that this  x value must be < 5........so we have

l 12x  + 5(1) - 60 l

_________________   =  1

sqrt ( 12^2 + 5^2 )

So we have that

12x - 55  =  13

12x = 68

x = 68/12   > 5.....so....no good

OR

12x - 55 = -13

12x = 42

x = 42/12  =  7/2  = 3.5  ......and this is what we need....this = the radius of the cross-section

So....we need to find  a parallel line to  y =  12x + 5y -60   and passing  through "H"  = (3.5, 1)

The side of our cross-section will lie on this line

So....the equation of this line will be

y = (-12/5)(x - 3.5) + 1

And letting x = 0....we can find the y coordinate of "G"  =  the distance from G to JB

So...we have.....

y = (-12/5) (-3.5) + 1

y = 9.4=  the distance from G to JB

However....we need to subtract 1 from this value because we actually want the distance that G is from KH

So...the height of the cross-section = 8.4

So....the radius of this cross-section =  3.5   and it's height = 8.4

So.....rotating this about the y axis produces the volume of the cross-section =

pi (3.5)^2 (8.4) / 3   =   34.3 pi   cm^3

And the ratio  of the volume of the  cross-section to the total  volume of the cone is

34.3 pi              34.3

______  =        _____     =   .343

100  pi               100   Mar 17, 2019
edited by CPhill  Mar 17, 2019
#2
+1

Thanks!!!

dgfgrafgdfge111  Mar 18, 2019