Question 1083682
 

 Large cone is{{{ r[1]=9cm}}} and {{{L[1]=15cm }}} slant height

small cone has a lateral area of {{{LA=60 pi}}}

The lateral area of a cone is

 {{{LA=pi * r * L}}} where {{{r}}} is the radius and {{{L}}} is the slant height

{{{60pi=pi * r * L}}}

{{{60= r * L}}}

{{{60/L= r }}}


since cones similar, radii and  slant heights are proportional:

{{{r[1]/r=L[1]/L}}}

{{{9/(60/L)=15/L}}}

{{{9L/60=15/L}}}

{{{3L/20=15/L}}}

{{{3L*L=15*20}}}

{{{L^2=(15*20)/3}}}

{{{L^2=5*20}}}

{{{L^2=100}}}

{{{L=10cm}}}->  slant height of a small cone


{{{60/L= r }}}

{{{60/10cm= r }}}

{{{6cm= r }}}-> radius of  a small cone


now we can find a volume:

{{{V=(1/3)b*h}}} where {{{b}}} is base and {{{h}}} height of the cone

base is a circle: area is {{{r^2pi}}}

so, {{{V=(1/3)r^2*pi*h}}}

we will find height using Pythagorean theorem: 

{{{h^2=L^2-r^2}}} ->{{{h=sqrt(10^2-6^2)}}} ->{{{h=sqrt(100-36)}}}->{{{h=sqrt(64)}}}->{{{h=8cm}}}

{{{V=(1/3)(6cm)^2*pi*8cm}}}

{{{V=(1/cross(3))(cross(36)12cm^2*pi*8cm)}}}

{{{V=12cm^2*pi*8cm}}}

{{{V=96cm^3*pi}}}